Majorana Collaboration

# Experimental study of 13C($\alpha,n)^{16}$O reactions in the Majorana
Demonstrator calibration data

I.J. Arnquist  Pacific Northwest National Laboratory, Richland, WA 99354, USA
F.T. Avignone III  Department of Physics and Astronomy, University of South
Carolina, Columbia, SC 29208, USA  Oak Ridge National Laboratory, Oak Ridge,
TN 37830, USA  A.S. Barabash  National Research Center “Kurchatov Institute”
Institute for Theoretical and Experimental Physics, Moscow, 117218 Russia
C.J. Barton  Department of Physics, University of South Dakota, Vermillion, SD
57069, USA  K.H. Bhimani  Department of Physics and Astronomy, University of
North Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  E. Blalock  Department of Physics, North
Carolina State University, Raleigh, NC 27695, USA  Triangle Universities
Nuclear Laboratory, Durham, NC 27708, USA  B. Bos  Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  M. Busch  Department
of Physics, Duke University, Durham, NC 27708, USA  Triangle Universities
Nuclear Laboratory, Durham, NC 27708, USA  M. Buuck  SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA  Center for Experimental Nuclear Physics
and Astrophysics, and Department of Physics, University of Washington,
Seattle, WA 98195, USA  T.S. Caldwell  Department of Physics and Astronomy,
University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  Y-D. Chan  Nuclear
Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720,
USA  C.D. Christofferson  South Dakota School of Mines and Technology, Rapid
City, SD 57701, USA  P.-H. Chu  Los Alamos National Laboratory, Los Alamos, NM
87545, USA  M.L. Clark  Department of Physics and Astronomy, University of
North Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  C. Cuesta  Centro de Investigaciones
Energéticas, Medioambientales y Tecnológicas, CIEMAT 28040, Madrid, Spain
J.A. Detwiler  Center for Experimental Nuclear Physics and Astrophysics, and
Department of Physics, University of Washington, Seattle, WA 98195, USA  Yu.
Efremenko  Department of Physics and Astronomy, University of Tennessee,
Knoxville, TN 37916, USA  Oak Ridge National Laboratory, Oak Ridge, TN 37830,
USA  H. Ejiri  Research Center for Nuclear Physics, Osaka University, Ibaraki,
Osaka 567-0047, Japan  S.R. Elliott  Los Alamos National Laboratory, Los
Alamos, NM 87545, USA  G.K. Giovanetti  Physics Department, Williams College,
Williamstown, MA 01267, USA  M.P. Green  Department of Physics, North Carolina
State University, Raleigh, NC 27695, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  Oak Ridge National Laboratory, Oak Ridge,
TN 37830, USA  J. Gruszko  Department of Physics and Astronomy, University of
North Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  I.S. Guinn  Department of Physics and
Astronomy, University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  V.E. Guiseppe  Oak
Ridge National Laboratory, Oak Ridge, TN 37830, USA  C.R. Haufe  Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514,
USA  Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  R.
Henning  Department of Physics and Astronomy, University of North Carolina,
Chapel Hill, NC 27514, USA  Triangle Universities Nuclear Laboratory, Durham,
NC 27708, USA  D. Hervas Aguilar  Department of Physics and Astronomy,
University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  E.W. Hoppe  Pacific
Northwest National Laboratory, Richland, WA 99354, USA  A. Hostiuc  Center for
Experimental Nuclear Physics and Astrophysics, and Department of Physics,
University of Washington, Seattle, WA 98195, USA  M.F. Kidd  Tennessee Tech
University, Cookeville, TN 38505, USA  I. Kim  Los Alamos National Laboratory,
Los Alamos, NM 87545, USA  R.T. Kouzes  Pacific Northwest National Laboratory,
Richland, WA 99354, USA  T.E. Lannen V  Department of Physics and Astronomy,
University of South Carolina, Columbia, SC 29208, USA  A. Li  Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27514,
USA  Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  A.M.
Lopez  Department of Physics and Astronomy, University of Tennessee,
Knoxville, TN 37916, USA  J.M. López-Castaño  Oak Ridge National Laboratory,
Oak Ridge, TN 37830, USA  E.L. Martin  Present address: Duke University,
Durham, NC 27708  Department of Physics and Astronomy, University of North
Carolina, Chapel Hill, NC 27514, USA  Triangle Universities Nuclear
Laboratory, Durham, NC 27708, USA  R.D. Martin  Department of Physics,
Engineering Physics and Astronomy, Queen’s University, Kingston, ON K7L 3N6,
Canada  R. Massarczyk  Los Alamos National Laboratory, Los Alamos, NM 87545,
USA  S.J. Meijer  Los Alamos National Laboratory, Los Alamos, NM 87545, USA
T.K. Oli<EMAIL_ADDRESS>Department of Physics, University of
South Dakota, Vermillion, SD 57069, USA  G. Othman  Present address:
Universität Hamburg, Institut für Experimentalphysik, Hamburg, Germany
Department of Physics and Astronomy, University of North Carolina, Chapel
Hill, NC 27514, USA  Triangle Universities Nuclear Laboratory, Durham, NC
27708, USA  L.S. Paudel  Department of Physics, University of South Dakota,
Vermillion, SD 57069, USA  W. Pettus  Department of Physics, Indiana
University, Bloomington, IN 47405, USA  IU Center for Exploration of Energy
and Matter, Bloomington, IN 47408, USA  A.W.P. Poon  Nuclear Science Division,
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  D.C. Radford
Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA  A.L. Reine
Department of Physics and Astronomy, University of North Carolina, Chapel
Hill, NC 27514, USA  Triangle Universities Nuclear Laboratory, Durham, NC
27708, USA  K. Rielage  Los Alamos National Laboratory, Los Alamos, NM 87545,
USA  N.W. Ruof  Center for Experimental Nuclear Physics and Astrophysics, and
Department of Physics, University of Washington, Seattle, WA 98195, USA  D.
Tedeschi  Department of Physics and Astronomy, University of South Carolina,
Columbia, SC 29208, USA  R.L. Varner  Oak Ridge National Laboratory, Oak
Ridge, TN 37830, USA  S. Vasilyev  Joint Institute for Nuclear Research,
Dubna, 141980 Russia  J.F. Wilkerson  Department of Physics and Astronomy,
University of North Carolina, Chapel Hill, NC 27514, USA  Triangle
Universities Nuclear Laboratory, Durham, NC 27708, USA  Oak Ridge National
Laboratory, Oak Ridge, TN 37830, USA  C. Wiseman  Center for Experimental
Nuclear Physics and Astrophysics, and Department of Physics, University of
Washington, Seattle, WA 98195, USA  W. Xu  Department of Physics, University
of South Dakota, Vermillion, SD 57069, USA  C.-H. Yu  Oak Ridge National
Laboratory, Oak Ridge, TN 37830, USA  B.X. Zhu  Present address: Jet
Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109,
USA  Los Alamos National Laboratory, Los Alamos, NM 87545, USA

###### Abstract

Neutron captures and delayed decays of reaction products are common sources of
backgrounds in ultra-rare event searches. In this work, we studied
13C($\alpha,n)^{16}$O reactions induced by $\alpha$-particles emitted within
the calibration sources of the Majorana Demonstrator. These sources are
thorium-based calibration standards enclosed in carbon-rich materials. The
reaction rate was estimated by using the 6129-keV $\gamma$-rays emitted from
the excited 16O states that are populated when the incoming $\alpha$-particles
exceed the reaction Q-value. Thanks to the excellent energy performance of the
Demonstrator’s germanium detectors, these characteristic photons can be
clearly observed in the calibration data. Facilitated by Geant4 simulations, a
comparison between the observed 6129-keV photon rates and predictions by a
TALYS-based software was performed. The measurements and predictions were
found to be consistent, albeit with large statistical uncertainties. This
agreement provides support for background projections from
($\alpha,n$)-reactions in future double-beta decay search efforts.

##  I Introduction

Neutron-related reactions are an important source of background in underground
neutrino and dark matter experiments [1, 2, 3, 4]. One common source of
neutrons is ($\alpha,n$) reactions. Neutrons may penetrate shielding layers
before being captured on sensitive detector materials, often creating
radioactive isotopes, the delayed decays of which could be difficult to reject
due to a lack of coincidence timing information. For example, in germanium-
based neutrinoless double-beta decay ($0\nu\beta\beta$) experiments, neutron
captures on 76Ge create 77Ge (half-life: 11.3 hr) and 77mGe (half-life: 53.7
s) isotopes. The $\beta$-decay of these isotopes could potentially produce
signals similar to $0\nu\beta\beta$ and with energies near the double-beta
decay Q-value (Qββ) of 76Ge. This background has been studied in detail [5,
6].

232Th and 238U decay chains contain several $\alpha$-emitters. These
naturally-occurring isotopes are present in detector materials, and various
$\alpha$-particles with energies up to 9 MeV are emitted, initiating a range
of $(\alpha,n)$ reactions. Even though the cleanest materials can be assayed
and selected [7], stringent background requirements, especially for future
experiments, demand an understanding of these neutron contributions with
reasonable detail and precision. In particular, different types of plastic
materials are widely used in low-background experiments, e.g. for electrical
insulation and neutron shielding. In these carbon-rich plastic materials, the
13C($\alpha,n)^{16}$O reaction is a major source of neutrons.

Besides its role as a background, the 13C($\alpha,n)^{16}$O reaction is
considered the most important neutron source for s-process nucleosynthesis in
low-mass asymptotic giant branch stars [8, 9, 10, 11, 12]. This reaction and
its cross section have been studied, and the results agree reasonably well
among different measurements for low-energy $\alpha$-particles below about 5
MeV [13, 14, 15, 16, 17, 18, 19]. At higher $\alpha$ energies, precise cross
section measurements are sparse [18, 20], although new studies have been
published for the 5-to-6 MeV region [4] and more measurements are planned in
the near future [19].

In addition to relying on measured data, one can obtain ($\alpha,n$) cross
sections from a statistical modeling approach using a nuclear reaction code
such as TALYS [21]. The TALYS-generated Evaluated Nuclear Data Libraries
(TENDL) merges the TALYS nuclear model with data available in the JENDL [22]
and ENDF [23] databases. In the case of 13C($\alpha,n)^{16}$O, TALYS can
predict partial cross sections of different reaction channels noted as
13C($\alpha,n_{j}$)16O, where $j$ identifies neutrons associated with
different states of 16O. The TALYS-generated cross sections as a function of
$\alpha$-particle energy are shown in Fig. 1. Although such a statistical
model can be imprecise when predicting the detailed resonance structure as
pointed out by Ref. [4], its overall agreement can be used to approximate the
reaction rate, allowing neutron background predictions for low-background
experiments. It is reasonable to use this approach especially when precise
measurements are sparse over the entire range of $\alpha$ energies relevant
for ($\alpha,n$) backgrounds, which is typically broad.

Figure 1: Total cross section and the partial cross sections for the
13C($\alpha,n)^{16}$O reactions as a function of incident $\alpha$-particle
energy available from the decay chain of 228Th. These cross sections are
generated by TALYS-1.95. The results of the new TALYS version are consistent
with branching ratios obtained from Ref. [24] that used TALYS-1.8.

In this paper, we report an analysis of several years of calibration data
taken by the Majorana Demonstrator experiment, which resulted in a measurement
of characteristic 6129-keV photons emitted following the
13C($\alpha,n_{2})^{16}$O reactions, where the second excited state (3-) of
16O is populated. We compare the measurement with a prediction from NeuCBOT
(Neutron Calculator Based On TALYS) [25, 26]. Section II of this paper
discusses the ($\alpha,n$) reactions within the calibration sources of the
Demonstrator. Section III introduces the experimental techniques and analysis
used to identify the 6129-keV photons. Section IV describes how the TALYS-
based NeuCBOT and a Geant4-based software for the Demonstrator are used to
predict the number of observable events. Section V discusses how the same
procedure can be used to estimate the background contribution to
$0\nu\beta\beta$ measurements. The last section shows how similar techniques
could play an essential role in future experiments with more stringent
background goals.

##  II  13C($\alpha,n$)16O reactions in Calibration

The Majorana Demonstrator experiment searched for $0\nu\beta\beta$ in 76Ge
using P-type Point Contact (PPC) High Purity Germanium (HPGe) detectors [27].
The Demonstrator was operated at the 4850-foot level of the Sanford
Underground Research Facility in Lead, South Dakota, with two modules of HPGe
detectors placed in an ultra-clean and heavily shielded environment as shown
in Fig 2. The HPGe detectors had a combined total mass of 44.1 kg, of which
29.7 kg was enriched to 88% in 76Ge with the rest being natural Ge. In March
2021, the Demonstrator completed its data-taking campaign with enriched
detectors and it continues taking data with natural detectors for background
studies and other physics. The Demonstrator’s HPGe detectors in combination
with low-noise electronics have achieved good linearity over a broad energy
range [28] and best-in-field energy resolution with a full width at half
maximum (FWHM) approaching 0.1% at the 2039 keV Qββ of 76Ge [29]. This
excellent energy performance coupled with the low energy threshold and low-
background of the Demonstrator makes it a competitive $0\nu\beta\beta$
experiment and allows for other physics beyond the Standard Model [30, 31, 32,
33, 34].

Figure 2: A schematic of the Majorana Demonstrator with two modules of HPGe
detectors surrounded with layers of shielding [30].

Ultra radiopure materials were used in the construction of the Demonstrator,
particularly in the vicinity of germanium detectors, which are placed inside
layers of compact shielding [35]. A weekly calibration is required to monitor
detector stability and provide data for developing analysis cuts. The thorium
isotope 228Th was selected as the calibration source because its decay chain
emits several $\gamma$-rays that span from a few hundred keV up to 2615 keV,
covering the Qββ of 76Ge and allowing for analysis over a wide energy range.
The Demonstrator’s calibration line sources were manufactured by Eckert &
Ziegler Analytics, Inc 111http://www.ezag.com/home/. Each line source is made
of thoriated epoxy encapsulated in a tube made of PTFE [36]. During
calibrations, the line source was deployed into the calibration track, which
surrounds the cryostat in a helical path [36], as shown in Fig 3.

Figure 3: A diagram that shows one module, the detector strings within, and
the calibration track (highlighted) through which a line source is deployed
during calibrations.

The $\gamma$-rays emitted within the thorium decay chain are used for
calibration and detector characterization. The decay chain ends when it
reaches 208Pb producing several $\alpha$-emitters along the way. Table 1 shows
the energies of the main $\alpha$-particles, which lie between 5.34 MeV and
8.79 MeV. When traversing the epoxy in the calibration source, an
$\alpha$-particle could initiate $(\alpha,n)$ reactions in 13C, 17O, 18O,
35Cl, and 37Cl, of which reactions with 13C dominate.
13C($\alpha,n_{2})^{16}$O reactions are possible with $\alpha$-particles above
about 5 MeV, resulting in 6129-keV photons. The Demonstrator’s excellent
energy performance allowed a clear observation of this 6129-keV signature on
top of the thorium photon energy spectrum during calibrations.

In 13C($\alpha,n$)16O reactions, an $\alpha$-particle is captured in 13C to
form the compound nucleus 17O∗, which decays to the ground state or excited
states of 16O by emitting a neutron. Figure 4 shows the simplified level
scheme of 16O that can be populated from the decay of 17O∗. Since the
$\alpha$-particles in the thorium chain have energy up to 8.79 MeV as listed
in Table 1, they can potentially open up the reactions channels of
($\alpha,n_{1}$), ($\alpha,n_{2}$), ($\alpha,n_{3}$), and ($\alpha,n_{4}$).
The population of the second excited state (3-) of 16O at 6129 keV is favored
for $\alpha$-particles with energy above 6 MeV, as shown by the turquoise line
in Fig. 1. The isomeric transition of the (3-) state to the ground state of
16O emits a characteristic 6129-keV photon, presenting a unique signature to
look for in calibration data. The 13C($\alpha,n_{2}$)16O reaction is described
in Eq. 1.

$\begin{split}{}^{13}\rm{C}+\alpha&\rightarrow^{17}\rm{O}^{*}\\\
&\rightarrow^{16}\rm{O}^{*}(3^{-})+n\\\
&\rightarrow^{16}\rm{O}(g.s.)+\gamma~{}(6129~{}\rm{keV})+n\end{split}$ (1)

Figure 4: The level scheme of 16O as populated in the 13C$(\alpha,n)^{16}$O
reaction (energy not to scale) simplified from Figure 1 of Ref. [4]. The
numerical index of the emitted neutrons n0, n1, n2 represents which final
state in 16O is populated. Due to selection rules, the 0+ (6049 keV) state
deexcites via the emission of an e+e- pair, while the 3- (6129 keV) state
deexcites through $\gamma$-ray emission. Data from [24, 4].

.

Table 1: Energies of primary $\alpha$-particles from the decay chain of 228Th. Energy data are taken from Nuclear structure & decay Data (NuDat 3.0)333https://www.nndc.bnl.gov/nudat3/. $\alpha$-particle energy (MeV) | Parent isotope |  Intensity (per 228Th decay)
---|---|---
5.423 |  228Th | 0.734
5.340 |  228Th | 0.260
5.685 |  224Ra | 0.949
5.449 |  224Ra | 0.051
6.288 |  220Rn | 0.999
6.778 |  216Po | 0.999
6.050 |  212Bi | 0.090
6.089 |  212Bi | 0.035
8.785 |  212Po | 0.641

##  III Analysis of data

For this work, data from weekly calibrations was analyzed in several steps,
including data selection and data quality checks, validation of the Geant4
simulation, and the signature search at higher energies. The Majorana
Demonstrator Data Acquisition (DAQ) system records waveforms from each HPGe
detector using two digitization channels with different amplifications, called
the low-gain and high-gain channels. The high-gain channels have been
extensively used for double-beta decay searches [29], but they saturate around
3 MeV due to the digitization range. The low-gain channels have a wider
dynamic range up to 10 MeV and allow a study of signatures with higher energy
depositions, e.g. by cosmic ray reactions or neutrons. The low-gain channels
are used here to search for the 6129-keV photons.

###  III.1  Data quality and simulation benchmarking

The modular approach of the Demonstrator enabled a flexible construction as
well as early data-taking once the first module was constructed. Each
calibration source was deployed separately for most of the Demonstrator’s
calibration data, except for a period after the installation of the second
module when two sources were deployed simultaneously to calibrate both
modules. For these calibrations, the DAQ throughput was potentially saturated.
Thus this analysis only uses data collected when one calibration source was
deployed at a time. Due to evolving calibration procedures, early
commissioning data are not used. For example, during commissioning, transition
runs during which the source was in motion were not flagged, which created
larger uncertainties in analysis time boundaries. The data analyzed here
include calibration data sets from the years 2016-2019, which were also used
in the analysis of the recent double-beta decay results [29]. The data quality
checks and channel selection used in the double-beta decay analysis [29] were
also applied here. Additional data quality checks based on the prominent
2615-keV $\gamma$-peak following the $\beta^{-}$ decay of 208Tl are applied to
the calibration data used in this analysis. If the 2615-keV, full energy event
rate in a run is found to deviate more than 3.5 $\sigma$ from the mean rate in
the same data set, the run is excluded from this analysis. Such deviations can
occur when, for example, the nitrogen dewars were filled, since the flow of
liquid nitrogen induced noise.

After the data quality checks, we compared the observed source activity
($A_{observed}$) with the expected activity ($A_{expected}$), defined as:

$A_{observed}=\frac{R}{\epsilon\times b}$ (2)

$A_{expected}=A_{0}e^{[-\lambda(t-t_{0})]}$ (3)

In Eq. 2, the observed activity of a calibration source during each weekly
calibration was estimated based on the rate, $R$, of the full energy 2615 keV
peak, the corresponding efficiency, $\epsilon$, of detecting the full energy
2615-keV photons, and the branching ratio, $b$, for the
212Bi$\rightarrow^{208}$Tl transition in the 228Th decay chain. The
Geant4-based [37] simulation package, MaGe [38], was used to estimate the
detection efficiency ($\epsilon$) of the 2615-keV photons originating from the
calibration sources in their deployed positions. In Eq. 3, the expected
activity of each calibration source is projected for every weekly calibration
based on the initial activity, $A_{0}$, reported by the vendor at a given time
$t_{0}$, the decay constant, $\lambda$, and the time of each calibration, $t$.
The decay chain is in equilibrium, so the decay constant is based on the
1.9-year half-life of 228Th. Uncertainties in the branching ratio, decay
constant, and calibration time are negligible, so the uncertainty in the
expected activity is dominated by the uncertainty in $A_{0}$, which was
$10.36\pm 0.60$ kBq on May 1, 2013.

As shown in Fig. 5, a good agreement was found between the expected and the
observed activity over multiple years of calibration data for both source
assemblies. This implies good accuracy for the simulations performed by MaGe
and gives confidence that MaGe can make correct efficiency predictions for the
analysis of the 6129 keV $\gamma$-rays.

Figure 5: Observed and expected activities for the two source assemblies used
in Majorana Demonstrator. The data points indicate the observed activity of
each source assembly for each weekly calibration, while the band represents
the expected activity which includes the vendor reported uncertainty. The
uncertainties in the observed activity are statistical only.

###  III.2 Signature search

The search for the 6129-keV photons from the 13C($\alpha,n_{2}$)16O reactions
was performed using the sum energy of events, which is obtained by summing all
coincident energy depositions over all active HPGe detectors within a 4 $\mu$s
window [30]. This sum energy is used because of the high probability for
several-MeV photons to distribute their full energy in multiple detectors.
Fig. 6 shows the sum energy spectrum above 1 MeV in calibration data. The
signature at 6129 keV following the 13C($\alpha,n_{2}$)16O reaction is clearly
visible. Fig. 7 provides a spectrum in a smaller energy band around the 6129
keV region. Events above 2615 keV are mostly due to summing, or random
coincidences of two unrelated decays in the calibration source. For the latter
one, the most prominent feature is the 5229-keV peak. When two 2615-keV
photons, the energy of which is 2614.511 keV, are in coincidence, the sum
energy appears to be twice of a photon energy. The zoomed-in plot of this peak
is shown in Fig. 8.

Figure 6: The sum energy spectrum of calibration data selected for the
analysis. It shows various $\gamma$-ray peaks, including 2615 keV, the
signature peak of 6129 keV, and other peaks from the calibration source and
peaks due to random coincidence events, and summing.  Figure 7: The signature
peak at 6129 keV from 13C($\alpha,n_{2}$)16O reactions in the Majorana
Demonstrator calibration sources, shown in blue color and fitted with Gaussian
in red. The gray-filled histogram is the peak shape from the simulation of 1
million 6129-keV photons from the calibration tracks.

We defined the region of interest (ROI) for the 6129-keV peak search as (6129
$\pm$ 10) keV based on the expected resolution in that energy region: about 2
keV ($1\sigma$) at 6 MeV, so the chosen window covers about 5$\sigma$ on each
side of the peak. A simple Gaussian fit to the signal peak found the mean to
be 6127$\pm$0.6 keV and the standard deviation to be 1.8$\pm$0.4 keV, as shown
in Fig. 7. A total of 9 events were found in the ROI in all data combined.
Given the low statistics, the uncertainties from the fit are relatively large
and less robust. As a cross-check, a simple Gaussian plus a flat background
was fit to the much stronger double coincidence 5229-keV peak in Fig. 8, where
the mean was found to be 5228$\pm$0.2 keV with a standard deviation of
2.0$\pm$0.1 keV. These full energy peaks are seen at their expected locations
and with their expected widths in the sum energy spectrum from the low-gain
channels, indicating a great energy performance extended to the energy range
of multiple-MeV.

Figure 8: The double coincidence peak at 5229 keV ($2615.5$ keV$\times 2$) in
the sum energy spectrum of the calibration data is fitted with with a Gaussian
plus a flat background.

As seen in Fig. 7, the signature peak at 6129 keV stands out clearly, so all
of the 9 observed events in the peak are considered to be the signal, i.e.
6129-keV photons following the 13C($\alpha,n_{2}$)16O reactions. Given that no
background events were found for at least 20 keV on both sides of the peak
outside the ROI, the potential background in the 40-keV region from 6099 keV
to 6159 keV excluding the 20-keV ROI can be determined as at most 1.29 counts
at a $1\sigma$ level, which translates to a $1\sigma$ upper limit of 0.64
counts of background in the ROI. To better determine the potential background
contribution, we also counted events in a much broader background region from
6 to 6.5 MeV, excluding the ROI around the 6129-keV peak. Based on 8 events in
this 480-keV background region, we projected the potential background to be
0.33 counts in the ROI. Incidentally, this projects 0.67 counts of background
from 6099 keV to 6159 keV excluding the 20-keV ROI, statistically consistent
with observing none, which would happen with a 50% probability. In short, the
observed number of signal events in the ROI in the combined data sets is 9,
while 0.33 counts is the estimated background contribution to the expected
number of events. The difference between 0.33 counts and 0.64 counts is
treated as a systematic uncertainty on the background contribution to the ROI.

##  IV  Comparisons with predictions

###  IV.1 Prediction calculation

NeuCBOT [25] is a software tool based on TALYS to calculate the neutron yield
and neutron energy spectra for ($\alpha,n$) reactions in materials. It models
the entire trajectory of $\alpha$-particles: initializing $\alpha$-particles
according to ENSDF evaluated nuclear decay data [39], tracking their energy
loss and range according to SRIM [40], and ultimately predicting the
($\alpha,n$) rate based on cross sections in TALYS-based TENDL. In this work,
the 6129-keV photon production rate was estimated by NeuCBOT using the partial
13C($\alpha,n_{2}$)16O cross sections from TALYS-1.95, and it was found to be
2.98 $\times 10^{-7}$ $\gamma$/Th-decay.

The detector configuration, such as the list of active detectors, can vary
over time. The source activity also reduces as thorium decays away. Therefore,
the number of predicted events was calculated for each weekly calibration and
summed together using:

$N=Y\times\sum_{i}A_{i}\times\epsilon_{i}\times T_{i}$ (4)

Here, $Y$ is the $\gamma$-ray production yield per decay of thorium, which is
constant for all data sets since the source assembly does not change. For each
weekly calibration $i$, the factors Ai, $\epsilon_{i}$, and Ti are the source
activity, detection efficiency for the 6129-keV photons, and live time,
respectively. The efficiency, $\epsilon_{i}$, was calculated with MaGe for the
6129-keV photon using the same geometry as for the 2615-keV analysis, but the
sum energy was used instead of individual detector energy for consistency. The
simulated 6129-keV peak shape in the sum energy spectrum for 6129-keV photons
uniformly seeded inside the calibration source is shown in Fig. 7. The same
ROI as in the data analysis was used to calculate the efficiency. Realistic
energy responses, including dead layer models of each detector [29], are
folded into the simulation, so the peak has slight deviations from Gaussian,
notably a low energy tail.

###  IV.2 Comparison of observed and expected events

Table 2 compares the expected number of 6129-keV $\gamma$-rays with the number
observed; the latter can be modelled by Poisson statistics with an unknown
true mean. Based on the observed signal counts, the confidence interval on the
mean of Poisson signals is calculated at a 90% confidence level (C.L.) using
the Feldman-Cousins statistical approach for small signals [41].

Sources of uncertainty in the expected counts are summarized in Table 3.
Uncertainties in the SRIM database are reported in Ref. [42]. Uncertainties
due to the chemical composition of the epoxy material and in the source
activities were both based on the specifications provided by the vendor. As
discussed before, the projected background contribution in the ROI depends on
the choice of background regions and the difference between the narrow 40-keV
and the wide 480-keV background regions is taken as the uncertainty. To assess
uncertainty associated with the calculation of the high energy photon
detection efficiency using the sum energy, we repeated the calibration source
activity analysis in Section III using the sum energy. On average, a 11.9%
difference in the source activity is observed between calculations based on
single detector energy and the sum energy at 2615-keV. The difference between
the vendor specification and the source activity based on the sum energy was
found to be smaller, so the 11.9% is an overestimation of the systematic
uncertainty in simulation.

Table 2: Expected and observed counts of 6129-keV photons. Expected counts are estimated based on Eq. 4, and the corresponding uncertainties are the combination of various uncertainties shown in Table 3. The range of signal mean is the 90% C.L. interval of Poisson signal mean based on observed signal counts in each data [41]. The individual data set labeling follows Demonstrator configuration changes as explained in Ref. [29]. Calibration Data Set |  Integrated Exposure Time (hour) | Expected Counts | Observed Counts |  90% Interval of Signal Mean given Observation
---|---|---|---|---
DS1 | 40.2 | 0.42$\pm$0.07 | 0 | [0.00, 2.44]
DS2 | 13.4 | 0.13$\pm$0.02 | 1 | [0.11, 4.36]
DS5 | 41.8 | 0.41$\pm$0.07 | 1 | [0.11, 4.36]
DS6a | 43.9 | 0.32$\pm$0.05 | 1 | [0.11, 4.36]
DS6b | 178.3 | 1.19$\pm$0.20 | 4 | [1.47, 8.60]
DS6c | 245.0 | 1.27$\pm$0.21 | 2 | [0.53, 5.91]
Total | 562.6 | 3.74$\pm$0.63 | 9 | [4.36, 15.30]
Table 3: Relative uncertainties for the expected number of counts. The total systematic uncertainty is the sum in quadrature of individual systematic contributions. $\gamma$ yield value due to uncertainties in the SRIM reported in [42] | 5.0%
---|---
Chemical composition in epoxy | 4.0%
Activity of the source as reported by Eckert & Ziegler | 5.8%
Systematic uncertainty in simulation | 11.9%
Statistical uncertainty in simulation | 1-2 % (neglected)
Systematic uncertainty in background contribution | 8.3%
Total systematic uncertainty | 16.9%

Figure 9 visualizes the comparison between the expected and the observed
number of 6129-keV photons. The observed number of events tends to be higher
than the expected number, however the statistical uncertainty in the
experimental data is large. The range of expected counts is overall consistent
with the 90% confidence interval on the observed signal strength. This
comparison suggests that TALYS cross sections combined with SRIM enables
reasonable estimations of ($\alpha,n$) rates. This consistency at a 90%
confidence level lends support to the approach of predicting neutron
production from $\alpha$-induced reactions in low-background experiments using
the presented tools.

Figure 9: Expected and observed number of 6129-keV photons with corresponding
uncertainties in each data set and in the combined data set. The error bars in
the observed counts indicate the 90% C.L. intervals on the mean of Poisson
signals as listed in Table 2.

##  V Background estimation for $0\nu\beta\beta$ search

Neutrons produced by ($\alpha,n$) reactions during the Demonstrator’s
calibration runs can enter the germanium crystals and get captured. After each
calibration, these sources were retracted to parked locations entirely outside
the shield. Therefore, only the neutrons produced during the calibration are
of concern. When 76Ge undergoes neutron capture, the ground state of 77Ge or
the metastable state, 77mGe, can be produced, both of which could $\beta$
decay with energy releases larger than the 2039 keV Qββ of $0\nu\beta\beta$ in
germanium [6]. The main background contributor here is the long-lived isotope
77Ge with a half-life of 11.2 hr, which can decay during the $0\nu\beta\beta$
decay data-taking periods following the hours-long calibration periods. The
metastable state 77mGe with a 54-second half-life is less of a concern.

Figure 10: Neutron energy spectrum from the ($\alpha,n$) reactions in the
epoxy. The spectrum is obtained by using NeuCBOT based on TALYS-1.95 generated
cross sections data.

Fig. 10 shows the NeuCBOT calculation of energies and yields of neutrons
generated from all types of ($\alpha$, n) reactions within the calibration
sources. MaGe was used to estimate the production and decay of 77Ge inside the
germanium crystals given this neutron flux. This background contribution was
estimated to be on the order of $10^{-5}$ cts/(keV-kg-year) before any
analysis cuts. This shows that calibration neutrons are a negligible
contribution compared with the total background measured in the Demonstrator
[29].

The GERDA experiment investigated a similar background source in their Phase I
data taking [43]. They estimated a background contribution of $10^{-4}$
cts/(keV-kg-year) for $0\nu\beta\beta$ by neutrons from calibration sources.
The higher background index can be explained by the stronger activity and
slightly different geometry used in GERDA. For GERDA Phase II data taking,
this background was minimized by deploying a new design of gold-encapsulated
thorium calibration source [43]. This design reduces the possible interaction
of $\alpha$-particles and it is adapted by the LEGEND calibration system [44].

Next-generation experiments searching for $0\nu\beta\beta$ have much more
stringent background requirements. Hence, potential background sources of
radiogenic ($\alpha,n$) neutrons from detector construction materials should
be examined carefully. While extensive efforts are in place to shield room and
cosmogenic neutrons, some neutron sources could be inside the water shielding
or are even introduced by shielding materials [6]. One example is the large
steel cryostat which houses the LEGEND main argon volume. The combination of
TALYS-based software can be valuable to provide rough estimations in these
cases, as investigated in Refs. [44, 45], using NeuCBOT in combination with
Geant4.

##  VI  Discussion and Summary

The work presented above combines the achievements of the Majorana
Demonstrator experiment in terms of excellent energy performance and robust
as-built simulations. The search for signatures in a wide energy range, well
beyond the Qββ of 76Ge, is possible due to excellent energy linearity and
resolution of the Majorana Demonstrator. These achievements result from the
intrinsic advantages of HPGe detectors in combination with low-noise
electronics and dedicated efforts on energy estimation corrections and
calibrations. The analysis presented here found a good agreement between
detected and expected energy for the signature at 6129 keV and verified the
algorithms at the sum peak of two 2615-keV $\gamma$-rays. We have shown that
the measured rate is consistent with simulations over various detector
configurations in multiple years of calibrations. The agreement between the
expected decay activity and the observed activity of the calibration sources
in the Majorana Demonstrator is reported for the first time, demonstrating the
excellent performance of the MaGe simulation software, which is also used by
GERDA and LEGEND.

Our work shows how signatures of ($\alpha,n$) reactions can be detected in
low-background experiments and how simulations are crucial in understanding
this radiogenic neutron background. The agreement between simulated and
measured rate is valuable feedback since ($\alpha,n$) data can be sparse, and
can have significant discrepancies, as pointed out by Ref. [4]. At 90% C.L.,
our measurement of the 6129-keV photons from the second excited state in 16O
is consistent with the predictions generated by the TALYS-based NeuCBOT
program, although the statistical uncertainty is large. This suggests that the
TALYS-based NeuCBOT provides a reasonable estimation of neutrons from thorium
impurities in carbon-rich organic materials. Our findings are widely
applicable, as thorium is one of the most common impurities, and carbon-rich
organic materials such as various plastics and epoxies are often used in
experiments in abundance. It is reasonable to expect that ($\alpha,n$)
reactions induced by alpha particles from thorium impurities in a range of
carbon-rich organic materials share similar profiles. While future experiments
may utilize materials with higher radiopurity than the current experiments,
the size and length of future experiments can result in a similar ($\alpha,n$)
background contribution for these rare event searches.

##  VII  Acknowledgements

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics under contract / award numbers
DE-AC02-05CH11231, DE-AC05-00OR22725, DE-AC05-76RL0130, DE-FG02-97ER41020, DE-
FG02-97ER41033, DE-FG02-97ER41041, DE-SC0012612, DE-SC0014445, DE-SC0018060,
and LANLEM77/LANLEM78. We acknowledge support from the Particle Astrophysics
Program and Nuclear Physics Program of the National Science Foundation through
grant numbers MRI-0923142, PHY-1003399, PHY-1102292, PHY-1206314, PHY-1614611,
PHY-1812409, PHY-1812356, and PHY-2111140. We gratefully acknowledge the
support of the Laboratory Directed Research & Development (LDRD) program at
Lawrence Berkeley National Laboratory for this work. We gratefully acknowledge
the support of the U.S. Department of Energy through the Los Alamos National
Laboratory LDRD Program and through the Pacific Northwest National Laboratory
LDRD Program for this work. We gratefully acknowledge the support of the South
Dakota Board of Regents Competitive Research Grant. We acknowledge support
from the Russian Foundation for Basic Research, grant No. 15-02-02919. We
acknowledge the support of the Natural Sciences and Engineering Research
Council of Canada, funding reference number SAPIN-2017-00023, and from the
Canada Foundation for Innovation John R. Evans Leaders Fund. This research
used resources provided by the Oak Ridge Leadership Computing Facility at Oak
Ridge National Laboratory and by the National Energy Research Scientific
Computing Center, a U.S. Department of Energy Office of Science User Facility.
We thank our hosts and colleagues at the Sanford Underground Research Facility
for their support.

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# Cavity-Induced Strong Magnon-Magnon Coupling in Altermagnets

Zhejunyu Jin  Huanhuan Yang  Zhaozhuo Zeng  Yunshan Cao  Peng Yan
<EMAIL_ADDRESS>School of Physics and State Key Laboratory of Electronic Thin
Films and Integrated Devices, University of Electronic Science and Technology
of China, Chengdu 610054, China

###### Abstract

Long-distance strong coupling between short-wavelength magnons remains an
outstanding challenge in quantum magnonics, an emerging interdiscipline
between magnonics and quantum information science. Recently, altermagnets are
identified as the third elementary class of magnets that break the time-
reversal symmetry without magnetization and thus combine characteristics of
conventional collinear ferromagnets and antiferromagnets. In this work, we
show that cavity photons can mediate the long-distance strong coupling of
exchange magnons with opposite chiralities in altermagnets, manifesting as an
anticrossing of the magnon-polariton spectrum in the extremely dispersive
regime. The predicted effective magnon-magnon coupling strongly depends on the
magnon propagation direction, and is thus highly anisotropic. Our findings are
intimately connected to the intrinsic nature of altermagnetic magnons, i.e.,
chirality-splitting-induced crossing of exchange magnons, which has no
counterpart in conventional ferromagnets or antiferromagnets, and may open a
new path way for magnon-based quantum information processing in altermagnets.

Introduction.—Magnons (quanta of spin waves) have been extensively explored
for wave-based sensing and computing concepts, due to their long lifetime and
high tunability Kruglyak2010 ; Chumak2015 ; Pirro2021 . The compatibility
between magnons and diverse quantum platforms such as qubit Tabuchi2015 ;
Mq2022 ; Mq2023 , phonon Agrawal2013 ; Streib2019 ; Bozhko2020 , and photon
Bai2011 ; Braggio2016; Harder2018  further amplifies the advantages of
utilizing magnons as a carrier for quantum information processing,
constituting quantum magnonics Yuan2022 . Coherent transfer of magnetic
information between two magnonic systems demands a strong magnon-magnon
coupling, which is usually generated by dipolar interaction Shiota2020 ,
interlayer exchange Chen2018 ; Ndiaye2017 ; Sklenar2021 , and in-plane
anisotropy Liensberger2019 , etc. However, limited by the effective range of
these magnetic interactions, they can only mediate the short-distance coupling
between magnons. It is noted that the coherent magnon-photon interaction has
been reported in hybrid cavity-magnet systems Soykal2010 ; Yuan2017 . Cavity
photons can also couple to two magnons simultaneously over a long distance,
inducing a nonlocal magnon-magnon interaction in both ferromagnets and
antiferromagnets Lambert2016 ; Rameshti2018 ; Grigoryan2019 ; Nair2022 ;
Zhang2023 ; Johansen2018 . Nevertheless, past studies focused on the long-
wavelength limits, i.e., magnetostatic magnons. The nonlocal strong coupling
between short-wavelength magnons remains an outstanding challenge in the
community.

Figure 1: Schematics of the cavity-induced long-distance magnon-magnon
coupling in altermagnets. The effective coupling (dashed green wavy line)
between two magnons with opposite chiralities is mediated by the cavity
photon.

Recently, an emerging class of magnets dubbed altermagnet was identified,
which maintains zero net macroscopic magnetization but exhibits a surprisingly
large spin-splitting Smejkal1 ; Smejkal2 ; Smejkal3 ; Mazin2023 ; Turek2022 ;
Feng2022 ; Ouassou2023 ; SZhang2023 ; Zhou2023 ; Hariki2023 ; Sun2023 ;
Bai2023 . These peculiar features are protected by the combined spin and
lattice symmetries, i.e., the spin-space inversion and crystallographic-space
rotation Smejkal1 ; Smejkal2 , which make altermagnets a promising platform
for quantum magnonics. It has been shown that the chiral-splitting effect
induces the crossing of infinitely-long-wavelength magnons (${\bf k}=0$, where
${\bf k}$ is the wavevector) with opposite chiralities Smejkal2 . But it
brings about two open issues for realizing strong nonlocal coupling between
exchange magnons: (i) How to push the crossing point to the short-wavelength
region; and (ii) How to realize the coupling between these two orthogonal
magnon modes, which was forbidden by angular momentum conservation, a
consequence of rotational invariance or isotropy.

In this Letter, we demonstrate the cavity-induced long-distance strong
coupling of exchange magnons in altermagnets (Fig. 1). We show that a
perpendicular magnetic field can lead to an upward (downward) shift of the
magnon branch with a low (high) group-velocity, resulting in an unavoided
level crossing at a finite wave number (${\bf k}\neq 0$). Then, we find that
the magnon degeneracy can be lifted by placing the altermagnet in a photonic
cavity, manifesting as an anticrossing in the magnon-polariton spectrum.
Surprisingly, it is observed that the photon frequency is orders of magnitude
higher than that of magnons at the anticrossing point, indicating a highly
dispersive magnon-photon coupling. By utilizing a second-order perturbation
theory, we derive the analytical formula of the cavity-induced effective
magnon-magnon coupling. We show that the indirect coupling strongly depends on
the magnon propagation direction due to the anisotropic nature of the exchange
interaction in altermagnets. Our results open the door for exploring quantum
magnonics based on the altermagnetism.

Chiral splitting of magnons in altermagnets.—Let us consider a two sublattice
altermagnet with anisotropic intralayer exchange interactions. The spin
Hamiltonian reads

$\displaystyle{\mathcal{H}}_{\text{alter}}=$
$\displaystyle-S^{2}\sum_{i,j}\big{[}J_{1}{\bf s}_{i,j}^{A}\cdot{\bf
s}_{i+1,j}^{A}+J_{2}{\bf s}_{i,j}^{B}\cdot{\bf s}_{i+1,j}^{B}$ (1)
$\displaystyle+J_{2}{\bf s}_{i,j}^{A}\cdot{\bf s}_{i,j+1}^{A}+J_{1}{\bf
s}_{i,j}^{B}\cdot{\bf s}_{i,j+1}^{B}+J_{3}{\bf s}_{i,j}^{A}\cdot{\bf
s}_{i,j}^{B}$ $\displaystyle+S^{-1}{\bf h}\cdot({\bf s}_{i,j}^{A}+{\bf
s}_{i,j}^{B})+K({\bf s}_{i,j}^{A}\cdot{\bf z})^{2}+K({\bf
s}_{i,j}^{B}\cdot{\bf z})^{2}\big{]},$

where $J_{1,2}>0$ represents the intralayer ferromagnetic exchange coupling
strength, $J_{3}<0$ is the interlayer antiferromagnetic exchange coupling
coefficient, ${\bf h}$ and $K$ denote the external magnetic field and the
magnetic anisotropy constant, respectively, ${\bf s}_{i,j}^{A}$ and ${\bf
s}_{i,j}^{B}$ are the normalized spins on sites $(i,j)$ of sublattices $A$ and
$B$, respectively, and $S$ is the length of the spin vector. Figure 2(a) shows
the crystal structure of a two-sublattice altermagnet. Under a combined
operation of two-fold spin-space rotation $\mathcal{C}_{s,2}$ (${\bf
s}^{A}\rightarrow{\bf s}^{B}$) and four-fold crystallographic-space rotation
$\mathcal{C}_{4}$ $(i\rightarrow j$ and $J_{1}\rightarrow J_{2})$, we find
that Hamiltonian (1) is invariant and thus respects the symmetry of
altermagnet Smejkal1 . We then obtain the dispersion relation of altermagnetic
magnons SM

$\displaystyle\omega_{{\bf k},\pm}$ $\displaystyle=\pm c_{1}({\bf
k})+c_{2}({\bf k}),$ (2)

with $c_{1}({\bf
k})=S^{2}\Big{\\{}h/S+(J_{1}-J_{2})\big{[}\cos(k_{y}a)-\cos(k_{x}a)\big{]}\Big{\\}}$
and $c_{2}({\bf
k})=S^{2}\Big{\\{}2K-(J_{2}+J_{1})\big{[}\cos(k_{x}a)+\cos(k_{y}a)-2\big{]}\Big{\\}}^{1/2}\Big{\\{}2K-2J_{3}-(J_{1}+J_{2})\big{[}\cos(k_{x}a)+\cos(k_{y}a)-2\big{]}\Big{\\}}^{1/2}$.
Here, $a$ is the lattice constant and $\pm$ corresponds to right-handed (RH)
and left-handed (LH) magnon modes, with respect to the $z$ axis, respectively.
In what follows, we use the following parameters to calculate the spectrum:
$J_{1}=0.4J_{2}$, $J_{3}=-1.25J_{2}$, $K=0.05J_{2}$, and $S=1$, if not stated
otherwise.

Figure 2: (a) Schematic illustration of a two-sublattice altermagnet. Spin-up
and-down atoms are labeled by red and blue spheres, respectively. The black
spheres correspond to nonmagnetic atoms. Magnon dispersion for $h/J_{2}=0$ (b)
and $h/J_{2}=0.3$ (c). Curves and circles represent the magnons propagating
along $x$ and $y$ direction, respectively. The handedness of magnons is
distinguished by contrast color (blue and red). Labels $x$ and $y$ represent
the propagation direction of magnons.

Different from the antiferromagnet, the degeneracy of RH and LH magnons is
broken even in the absence of the external magnetic field, resulting in the
unequal magnon group velocity, as shown in Fig. 2(b). In addition, it is noted
that the spectrum of RH magnons propagating along the $x$ axis is identical to
LH magnons propagating along the $y$ axis, and vice versa. When a
perpendicular magnetic field is applied, the energy degeneracy at $k=0$ is
removed ($k=|{\bf k}|$). However, we observe that, for magnons propagating
along $x$-direction, the branch with a lower group-velocity shifts upward,
while the one with a higher group-velocity shifts downward. This results in an
unavoided level crossing at a finite wave number, i.e., $k\neq 0$ [see red and
blue curves in Fig. 2(c)]. Such feature does not exist for magnons propagating
along the $y$ direction, which is an indication of the crossing anisotropy and
will be discussed below. A level crossing usually means the absence of
coupling. Particularly for two magnons with opposite handedness, their direct
coupling was thought to be forbidden by angular momentum conservation, a
consequence of rotational invariance. As to the long-distance coupling between
exchange magnons, it even looks like a dilemma at first sight because the
exchange interaction is extremely short-ranged. Next, we tackle this problem
in cavities.

Cavity-induced magnon-magnon coupling.—To this end, we consider the
altermagnet embedded in a photonic cavity, described by following Hamiltonian

$\displaystyle\mathcal{H}$
$\displaystyle={\mathcal{H}}_{\text{alter}}+{\mathcal{H}}_{\text{ph}}+{\mathcal{H}}_{\text{int}},$
(3) $\displaystyle{\mathcal{H}}_{\text{ph}}$
$\displaystyle=\frac{1}{2}\int(\epsilon_{0}{\bf E}^{2}+\frac{{\bf
B}^{2}}{\mu_{0}})d{\bf r},$ $\displaystyle{\mathcal{H}}_{\text{int}}$
$\displaystyle=-\frac{1}{\mu_{0}}\sum_{i,j}({\bf s}_{i,j}^{A}+{\bf
s}_{i,j}^{B})\cdot{\bf B},$

where $\mathcal{H}_{\text{ph}}$ is the photon Hamiltonian,
$\mathcal{H}_{\text{int}}$ is the magnon-photon coupling, ${\bf E}$ and ${\bf
B}$ are the electric and magnetic components of the electromagnetic wave,
respectively, and $\epsilon_{0}$ and $\mu_{0}$ are vacuum permittivity and
susceptibility, respectively. By using the Holstein-Primakoff transformation
HP :
$s^{A,+}=\sqrt{2}a,s^{B,+}=\sqrt{2}b^{\dagger},s^{A,-}=\sqrt{2}a^{\dagger},s^{B,-}=\sqrt{2}b,s^{A}_{z}=1-a^{\dagger}a,$
and $s^{B}_{z}=b^{\dagger}b-1$,

Figure 3: Dispersion relations of the magnon-polariton propagating along
$y$-axis (a,b) and $x$-axis (c,d) with zero (a,c) and finite magnetic field
(b,d). The dashed red, blue, and black lines represent uncoupled magnon modes
and cavity mode, while the silver-gray lines represent the magnon-polariton.
Insets show the details of the anticrossing in the spectrum. (e) Diagram of
energy level and interaction scheme for two magnons with frequencies
$\omega_{{\bf k},\pm}$ coupled to a cavity photon with frequency $\omega_{c}$,
by exchanging virtual photons.

where the spin operators $s^{m,\pm}=s^{m}_{x}\pm is^{m}_{y}$ with $m=A,B$, and
$a(b)$ and $a^{\dagger}(b^{\dagger})$ are the magnon annihilation and creation
operators for sublattice $A(B)$, respectively, we can express the magnon
Hamiltonian in the momentum space as

$\displaystyle\mathcal{H}_{\text{alter}}=$ $\displaystyle\sum_{{\bf
k}}\Big{\\{}\big{[}2J_{1}\cos(k_{x}a)+2J_{2}\cos(k_{y}a)+J_{3}-h$ (4)
$\displaystyle-2J_{2}-2J_{1}+2K\big{]}a_{\bf k}^{\dagger}a_{\bf
k}+\big{[}2J_{2}\cos(k_{x}a)$
$\displaystyle+2J_{1}\cos(k_{y}a)+J_{3}+h-2J_{2}-2J_{1}$
$\displaystyle+2K\big{]}b_{\bf k}^{\dagger}b_{\bf k}+J_{3}(a_{\bf k}b_{\bf
k}+a_{\bf k}^{{\dagger}}b_{\bf k}^{\dagger})\Big{\\}}.$

Here, we have assumed circularly-polarized photons, with the resulting photon
Hamiltonian and magnon-photon coupling as follows

$\displaystyle\mathcal{H}_{\text{ph}}$ $\displaystyle=\sum_{{\bf
k}}\omega_{c}(c_{\bf k}^{\dagger}c_{\bf k}+\frac{1}{2}),$ (5)
$\displaystyle\mathcal{H}_{\text{int}}$ $\displaystyle=\sum_{{\bf
k}}g_{c}(c_{\bf k}a_{\bf k}+c_{\bf k}^{\dagger}a_{\bf k}^{\dagger}+c_{\bf
k}b_{\bf k}^{\dagger}+c_{\bf k}^{\dagger}b_{\bf k}),$

where $\omega_{c}=v_{c}|{\bf k}|$ is the photon dispersion relation with
$v_{c}$ the speed of light and $g_{c}=\sqrt{\omega_{c}\mu_{0}N/2V}$ represents
the magnon-photon coupling with the number of spins $N$ and the volume of
cavity $V$ Yuan2017 . By utilizing the Bogoliubov transformation

$\displaystyle a_{\bf k}$ $\displaystyle=u_{\bf k}\alpha_{\bf k}+v_{\bf
k}\beta_{\bf k}^{\dagger},$ (6) $\displaystyle b_{\bf k}$
$\displaystyle=u_{\bf k}\beta_{\bf k}+v_{\bf k}\alpha_{\bf k}^{\dagger},$

where $u_{\bf k}=\sqrt{(\Delta_{\bf k}+1)/2}$, $v_{\bf k}=-\sqrt{(\Delta_{\bf
k}-1)/2}$, and

$\Delta_{\bf
k}=\sqrt{\frac{1}{1-\Big{\\{}\frac{J_{3}}{(J_{1}+J_{2})[\cos(k_{x}a)+\cos(k_{y}a)]+J_{3}-2(J_{1}+J_{2})-2K}\Big{\\}}^{2}}},$
(7)

the magnon Hamiltonian (4) can be diagonalized as

$\displaystyle\mathcal{H}_{\text{alter}}$ $\displaystyle=\sum_{{\bf
k}}(\omega_{{\bf k},-}\alpha_{\bf k}^{\dagger}\alpha_{\bf k}+\omega_{{\bf
k},+}\beta_{\bf k}^{\dagger}\beta_{\bf k}).$ (8)

The total Hamiltonian then can be recast as
$\mathcal{H}=\psi^{\dagger}\mathcal{M}\psi$ with the vector $\psi=(\alpha_{\bf
k},\beta_{\bf k}^{\dagger},c_{\bf k}^{\dagger})^{\dagger}$ and matrix

$\mathcal{M}=\left(\begin{array}[]{ccc}\omega_{{\bf k},-}&0&\frac{\lambda_{\bf
k}}{2}\\\ 0&\omega_{{\bf k},+}&\frac{\lambda_{\bf k}}{2}\\\ \frac{\lambda_{\bf
k}}{2}&\frac{\lambda_{\bf k}}{2}&\omega_{c}\\\ \end{array}\right),$ (9)

with $\lambda_{\bf k}=2g_{c}(u_{\bf k}+v_{\bf k})$, leading to the following
secular equation

$\displaystyle 4\omega^{3}-4(\omega_{{\bf k},+}+\omega_{{\bf
k},-}+\omega_{c})\omega^{2}+\lambda_{\bf k}^{2}(\omega_{{\bf
k},+}+\omega_{{\bf k},-})-4\omega_{{\bf k},+}\omega_{{\bf k},-}\omega_{c}$
(10) $\displaystyle+2\big{[}-\lambda_{\bf k}^{2}+2\omega_{{\bf
k},+}\omega_{{\bf k},-}+2(\omega_{{\bf k},+}+\omega_{{\bf
k},-})\omega_{c}\big{]}\omega=0.$

By numerical solving Eq. (10), one obtains the dispersion relation of the
coupled cavity-altermagnet. For the case without the magnetic field, near
${\bf k}=0$, one of the degenerated magnon bands is a dark mode that does not
interact with the cavity photon, while the other one does, see Figs. 3(a) and
(c). When the external field is present, the double degeneracy of magnons is
removed, and both magnon modes couple with the cavity photon, see Figs. 3(b)
and (d). These features are close to the case in antiferromagnets due to the
similar magnon dispersion in the long-wavelength limit Yuan2017 ; Yuan2020 .
Strikingly unlike the antiferromagnet, we find an anticrossing gap emerging at
$k_{x}=a^{-1}\arccos{\big{[}1-h/(J_{2}-J_{1})\big{]}}$ for magnons propagating
along $x$ direction. This gap represents the effective magnon-magnon coupling.
However, at this point, the photon frequency is much higher than magnon
($\omega_{c}/\omega_{{\bf k},\pm}>10^{5}$) [see Fig. 3(d)]. The magnon-photon
coupling is thus highly dispersive, where a resonant energy exchange between
magnon and photon is not allowed. But magnons can exchange energy through
virtual photons with the cavity, resulting in the indirect magnon-magnon
coupling, as illustrated in Fig. 3(e). To obtain an analytical understanding,
we derive the effective magnon-magnon coupling by using the following unitary
transformation

Figure 4: Analytical formula (red curves) and numerical results (black
circles) of the effective magnon-magnon coupling $g_{\rm{eff}}$ as a function
of the magnon-photon coupling strength $g=g_{c}/\sqrt{\omega_{c}/J_{2}}$ (a),
magnetic filed $h$ (b), exchange ratio $J_{1}/J_{2}$ (c), and anisotropy
constant $K$ (d).

$\displaystyle U=\exp\Big{[}$ $\displaystyle\frac{\lambda_{\bf
k}}{{2{\Delta_{1}}}}(c_{\bf k}^{\dagger}{\beta_{\bf k}}-\beta_{\bf
k}^{\dagger}{c_{\bf k}}+c_{\bf k}^{\dagger}{\alpha_{\bf k}}-\alpha_{\bf
k}^{\dagger}{c_{\bf k}})$ (11) $\displaystyle+\frac{\lambda_{\bf
k}}{{2{\Delta_{2}}}}(c_{\bf k}^{\dagger}\alpha_{\bf k}^{\dagger}-{c_{\bf
k}}{\alpha_{\bf k}}+c_{\bf k}^{\dagger}\beta_{\bf k}^{\dagger}-{c_{\bf
k}}{\beta_{\bf k}})\Big{]},$

and expanding the transformed Hamiltonian to the second order of $\lambda_{\bf
k}$

$\displaystyle\mathcal{H}_{\rm{eff}}$
$\displaystyle=U^{\dagger}\mathcal{H}U=\Big{[}(\omega^{\prime}+\frac{{{\lambda_{\bf
k}^{2}}}}{{4{\Delta_{2}}}}+\frac{{{\lambda_{\bf
k}^{2}}}}{{4{\Delta_{1}}}})\alpha_{\bf k}^{\dagger}{\alpha_{\bf
k}}+(\omega^{\prime}+\frac{{3{\lambda_{\bf k}^{2}}}}{{4{\Delta_{1}}}}$ (12)
$\displaystyle-\frac{{{\lambda_{\bf k}^{2}}}}{{4{\Delta_{2}}}})\beta_{\bf
k}^{\dagger}{\beta_{\bf k}}+({\omega_{c}}-\frac{{{\lambda_{\bf
k}^{2}}}}{{{\Delta_{1}}}})c_{\bf k}^{\dagger}{c_{\bf
k}}\Big{]}+g_{\rm{eff}}({\alpha_{\bf k}}\beta_{\bf k}^{\dagger}+{\beta_{\bf
k}}\alpha_{\bf k}^{\dagger}),$

where $\Delta_{1}=\omega_{c}-\omega^{\prime}$ and
$\Delta_{2}=-\omega_{c}-\omega^{\prime}$ with the magnon frequency
$\omega^{\prime}$ at the crossing point. Here,
$g_{\rm{eff}}=\frac{{{\lambda_{\bf k}^{2}}}}{{2{\Delta_{1}}}}$ describes the
effective magnon-magnon coupling mediated by virtual photons. In the
dispersive limit, a careful examination reveals that one of the hybridization
modes passes through the degeneracy point where
$\omega_{+}=\omega_{-}=\omega^{\prime}$, in sharp contrast to the observations
in resonantly coupled systems Xu2019 ; Li2022 . This feature can also be seen
from the three eigenvectors $\alpha_{\bf k}^{\dagger}-\beta_{\bf k},c_{\bf
k}-\frac{\sqrt{2}}{2}(\alpha_{\bf k}^{\dagger}+\beta_{\bf k})$, and $c_{\bf
k}+\frac{\sqrt{2}}{2}(\alpha_{\bf k}^{\dagger}+\beta_{\bf k})$, in which the
state $\alpha_{\bf k}^{\dagger}-\beta_{\bf k}$ is a magnon dark mode (also
known as a subradiant mode) that is decoupled from the cavity Zhang2015 .

Figure 4(a) demonstrates that the effective coupling induced by virtual photon
exchange is dominated by the second-order of the dispersive coupling
$\lambda_{\bf k}$. Since $k_{x}$ at the anticrossing point is a nonlinear
function of both the magnetic field $h$ and exchange ratio $J_{1}/J_{2}$, it
indicates a nonlinear dependence of the effective coupling strength
$g_{\rm{eff}}$ on $h$ and $J_{1}/J_{2}$, as shown in Figs. 4(b) and 4(c),
respectively. The anisotropy constant $K$ merely modifies the magnon
frequency, and hardly affects the coupling strength $g_{\rm{eff}}$ due to the
huge frequency mismatch between magnons and cavity photons, as shown in Fig.
4(d). Our theory well explains the numerical calculations.

Figure 5: (a) The wave vector $k_{c}$ at crossing point as a function $\phi$
for different magnetic fields $h$. (b) The analytical (red curve) and
numerical (black circles) results of the effective magnon-magnon coupling
$g_{\rm{eff}}$ as a function of $\phi$. In the calculations, we set
$J_{1}=0.4J_{2}$, $J_{3}=-1.25J_{2}$, and $K=0.05J_{2}$.

In the above calculations, we focused on the case of magnons propagating along
the $x$ direction. Notably, the anisotropic exchange interaction in
altermagnets is expected to generate anisotropic magnon-magnon couplings. To
illustrate this point, we express the magnon wavevector as
$k(\cos\phi\hat{x}+\sin\phi\hat{y})$ with $\phi$ being the polar angle. We
then derive the relation between the wavevector $k_{c}$ of the crossing point
and the angle $\phi$,

$\displaystyle\cos(k_{c}a\cos\phi)-\cos(k_{c}a\sin\phi)$
$\displaystyle=\frac{h}{J_{1}-J_{2}},$ (13)

which is a transcendental equation and it can only be solved numerically.

As plotted in Fig. 5(a), we observe that $k_{c}$ is symmetric about $\phi=0$
and $\pi$ due to the mirror symmetry of the $y-z$ plane, while it takes the
minimum when $\phi$ equals $0$ or $\pi$ because the difference in the group
velocities of magnons with opposite chiralities reaches the maximum in such
cases. When $\phi$ deviates from these two values, $k_{c}$ substantially
increases. In addition, it is noted that Eq. (13) has real solutions only when
the magnon propagation angle $\phi$ falls into two windows
$[-\phi_{0},\phi_{0}]$ and $[\pi-\phi_{0},\pi+\phi_{0}]$ with
$\phi_{0}\approx\sqrt{4+2h/(J_{1}-J_{2})}/\pi$, around $\phi=0$ and $\pi$.
Figure 5(a) also shows that the crossing point moves away from the origin as
the magnetic field increases. A direct consequence of the $\phi$-dependence of
$k_{c}$ is the anisotropy of the effective coupling $g_{\rm{eff}}$, as shown
in Fig. 5(b). Numerical results compares very well with our analytical
formula.

Discussion.—Using the magnetic parameters of $\rm{RuO_{2}}$: $J_{1}=0.8$ meV,
$J_{2}=2$ meV, $J_{3}=-3$ meV, and $K=0.1$ meV Smejkal2 , one can estimate the
effective magnon-magnon couping as $g_{\rm{eff}}=0.49$ meV for the spin
density $N/V$ $\sim 10^{23}$ cm-3 and $h=0.6$ meV. The resulting
cooperativity, i.e., the ratio of the indirect coupling to the magnon
dissipation rate, can reach $100$ for the Gilbert damping constant
$\alpha=1.0\times 10^{-3}$, indicating a strong coupling in the highly
dispersive region. It is noted that the dipole-dipole interaction can break
the degeneracy of magnons with opposite chiralities, while the resulting
direct magnon-magnon coupling is negligibly small compared to the cavity-
induced indirect coupling between exchange magnons Sheng2020 ; Tacchi2019 .
Its contribution can be further dismissed when considering the long-distance
coupling since the dipolar field generally decays as $1/d^{3}$ with $d$ the
spatial distance of two magnets.

To summarize, we have predicted the cavity mediated long-range interaction
between short-wavelength magnons in altermagnets. We showed that a
perpendicular magnetic field can shift the magnon degeneracy from the origin
to the exchange region, due to the unique chiral magnon splitting in
altermagnets. The strong long-distance coupling between magnons with opposite
handedness in cavities manifests as an anticrossing of the magnon-polariton
spectrum in the highly dispersive region. We derived the analytical formula of
the effective magnon-magnon coupling mediated by virtual photons and showed
that it is highly anisotropic. Our results may open a new pathway for
developing quantum magnonics and enrich the emerging research landscape of
altermagnetism.

###### Acknowledgements.

This work was funded by the National Key Research and Development Program
under Contract No. 2022YFA1402802 and the National Natural Science Foundation
of China under Grant No. 12074057.

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