# Strong magnon-photon coupling with chip-integrated YIG in the zero-
temperature limit

Paul G. Baity<EMAIL_ADDRESS>James Watt School of Engineering,
Electronics & Nanoscale Engineering Division, University of Glasgow, Glasgow
G12 8QQ, United Kingdom  Dmytro A. Bozhko  Center for Magnetism and Magnetic
Materials, Department of Physics and Energy Science, University of Colorado
Colorado Springs, Colorado Springs, Colorado 80918, USA  Rair Macêdo  James
Watt School of Engineering, Electronics & Nanoscale Engineering Division,
University of Glasgow, Glasgow G12 8QQ, United Kingdom  William Smith  SUPA,
School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ,
United Kingdom  Rory C. Holland  James Watt School of Engineering, Electronics
& Nanoscale Engineering Division, University of Glasgow, Glasgow G12 8QQ,
United Kingdom  Sergey Danilin  James Watt School of Engineering, Electronics
& Nanoscale Engineering Division, University of Glasgow, Glasgow G12 8QQ,
United Kingdom  Valentino Seferai  James Watt School of Engineering,
Electronics & Nanoscale Engineering Division, University of Glasgow, Glasgow
G12 8QQ, United Kingdom  João Barbosa  James Watt School of Engineering,
Electronics & Nanoscale Engineering Division, University of Glasgow, Glasgow
G12 8QQ, United Kingdom  Renju R. Peroor  Center for Magnetism and Magnetic
Materials, Department of Physics and Energy Science, University of Colorado
Colorado Springs, Colorado Springs, Colorado 80918, USA  Sara Goldman  Center
for Magnetism and Magnetic Materials, Department of Physics and Energy
Science, University of Colorado Colorado Springs, Colorado Springs, Colorado
80918, USA  Umberto Nasti  James Watt School of Engineering, Electronics &
Nanoscale Engineering Division, University of Glasgow, Glasgow G12 8QQ, United
Kingdom  Current affiliation: School of Engineering & Physical Sciences,
Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom  Jharna Paul  James
Watt School of Engineering, Electronics & Nanoscale Engineering Division,
University of Glasgow, Glasgow G12 8QQ, United Kingdom  Robert H. Hadfield
James Watt School of Engineering, Electronics & Nanoscale Engineering
Division, University of Glasgow, Glasgow G12 8QQ, United Kingdom  Stephen
McVitie  SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow
G12 8QQ, United Kingdom  Martin Weides  James Watt School of Engineering,
Electronics & Nanoscale Engineering Division, University of Glasgow, Glasgow
G12 8QQ, United Kingdom

###### Abstract

The cross-integration of spin-wave and superconducting technologies is a
promising method for creating novel hybrid devices for future information
processing technologies to store, manipulate, or convert data in both
classical and quantum regimes. Hybrid magnon-polariton systems have been
widely studied using bulk Yttrium Iron Garnet (Y3Fe5O12, YIG) and three-
dimensional microwave photon cavities. However, limitations in YIG growth have
thus far prevented its incorporation into CMOS compatible technology such as
high quality factor superconducting quantum technology. To overcome this
impediment, we have used Plasma Focused Ion Beam (PFIB) technology—taking
advantage of precision placement down to the micron-scale—to integrate YIG
with superconducting microwave devices. Ferromagnetic resonance has been
measured at milliKelvin temperatures on PFIB-processed YIG samples using
planar microwave circuits. Furthermore, we demonstrate strong coupling between
superconducting resonator and YIG ferromagnetic resonance modes by maintaining
reasonably low loss while reducing the system down to the micron scale. This
achievement of strong coupling on-chip is a crucial step toward fabrication of
functional hybrid quantum devices that advantage from spin-wave and
superconducting components.

††preprint: AIP/123-QED Figure 1: SEM images showing the PFIB processing steps
for YIG. (a) Section of a YIG film that has been milled out using a Xe ion
beam. An upside down pyramid structure with an approximate $50\times 50$
$\mu$m2 base is milled from the surface of the YIG film. (b) The pyramid is
attached to a nanomanipulator via Pt deposition on the manipulator tip.
Afterward, the connecting bridge (seen in (a)) is milled away and the pyramid
is extracted from the film surface as shown. (c) The extracted pyramid from
(b) is attached to a metal finger (right object) using Pt deposition. From
this position, the pyramid is milled flat into a cuboid shape. The cuboid can
be cut into smaller sections (e.g. $50\times 10\times 5$ $\mu$m3). (d) The
samples are then placed onto the NbN microwave devices using the
nanomanipulator. The YIG samples are affixed to the device surfaces by
precision Pt deposition. The YIG is milled free from the nanomanipulator by
milling briefly near the area connecting the nanomanipulator and YIG sample.
(e) An SEM image of NbN lumped element resonators with YIG placed onto the
central inductive line where the magnetic antinode is. The YIG samples are
oriented with respect to the shown axes. (f) Simulation of the NbN resonator
with zero-field frequency $\omega_{p0}/2\pi=12.517$ GHz. The field profiles
were determined for the bare resonator without YIG and simulated using COMSOL
Multiphysics® software.

In recent years, there has been large investment into research on cavity
magnon-polariton systems with strongly coupled spin-wave and photonic
componentsHuebl  _et al._ (2013); Tabuchi  _et al._ (2014); Zhang  _et al._
(2014, 2015); Morris  _et al._ (2017); Boventer  _et al._ (2018, 2020);
Trempler  _et al._ (2020). This research is driven in part by the promise of
creating new hybrid devices using spin-wave and other technologies, such as
superconducting quantum technologyTabuchi  _et al._ (2015). Indeed, hybrid
ferromagnetic-superconducting systems have been realized using both a 3D
electromagnetic cavityTabuchi  _et al._ (2015) and planarized 2D resonatorsLi
_et al._ (2019). This latter work made a large step toward full integration of
spin-wave and superconducting technologies, however, the large magnon line-
widths of ferromagnetic metals, such as permalloy (Ni80Fe20), pose deleterious
constraints on quantum coherent applications. The ferrimagnetic insulator
yttrium iron garnet (Y3Fe5O12, YIG) on the other hand, has a spin-wave damping
which is orders of magnitude lower that metallic ferromagnets as well as much
longer magnon lifetimesCherepanov, Kolokolov, and L’vov (1993); Kosen  _et
al._ (2019); Mihalceanu  _et al._ (2018). Therefore, this material has been
the forefront candidate for several applications in the field of spin-wave
technology (aka magnonicsKruglyak, Demokritov, and Grundler (2010); Serga,
Chumak, and Hillebrands (2010); Lenk  _et al._ (2011); Fischer  _et al._
(2017); Frey  _et al._ (2020); Prokopenko  _et al._ (2019)). However, despite
its exceptional spin-wave qualities, YIG poses issues of its own, specifically
its difficult growth and incompatibility standard processing techniques. The
growth of high-quality YIG typically requires specialized substrates such as
gadolinium gallium garnet (Gd3Ga5O12, GGG), making it incompatible with CMOS
processing techniquesLenk  _et al._ (2011). Here we present a method for
integrating high-quality YIG with superconducting quantum technologies using
plasma focused ion beam (PFIB) technologies. Our primary result is the
achievement of strong coupling on chip devices comprising a superconducting
resonator and a YIG sample at sub-Kelvin temperatures.

Sample |  $x$ [$\mu$m] |  $y$ [$\mu$m] |  $z$ [$\mu$m] | $D_{x}$ | $D_{y}$ | $D_{z}$
---|---|---|---|---|---|---
# 1 | $50\pm 6$ | $15\pm 1$ | $45.8\pm 0.3$ | $0.187\pm 0.004$ | $0.61\pm 0.01$ | $0.204\pm 0.002$
# 2 | $10.8\pm 0.5$ | $6.1\pm 0.3$ | $36.3\pm 0.4$ | $0.335\pm 0.004$ | $0.570\pm 0.007$ | $0.0958\pm 0.0003$
# 3 | $12.5\pm 0.5$ | $6.5\pm 0.5$ | $47.5\pm 0.5$ | $0.326\pm 0.005$ | $0.59\pm 0.01$ | $0.0818\pm 0.0003$
Table 1:  Sample dimensions and their calculated demagnetisation factors.
Errors on sample dimensions correspond to the deviations from the cuboid shape
across each sample. The demagnetization factors were calculated from sample
dimensions using equations in Ref. Aharoni, 1998. Errors on the
demagnetization factors reflect the shape errors. For the sample dimensions,
the $z$ direction is oriented parallel to the applied field, along the device
transmission line length (see insets of Figs. 2(a) and (b)). The $x$ and $y$
directions are oriented to the respective in-plane and out-of-plane
perpendiculars to $z$ axis.

The hybrid superconducting and ferromagnetic devices in this study were
fabricated in two stages: the fabrication of planar superconducting devices
and the integration of PFIB-processed YIG onto those devices. Superconducting
planar devices were fabricated from 100 nm thick niobium nitrate (NbN) films.
The films were deposited onto intrinsic silicon substrates at room temperature
using reactive sputtering of Nb in a mixture of N2 and Ar gas flow in a
similar fashion to other studiesBanerjee  _et al._ (2018). NbN material was
chosen for its high critical field values, which ensures that the devices
remain superconducting and operable under applied field. The superconducting
transition temperature was found to be $\sim 11$ K using four-probe
resistivity measurements. Planar devices were fabricated from the NbN films
using photo- and electron-beam lithography and reactive ion etching methods.

YIG samples were processed from a 100 $\mu$m thick film, grown on GGG, using
Xe plasma focused ion beam. Processing steps for the YIG samples are shown in
Fig. 1(a-d). First, a large section of the film is first milled out using a
wide beam column. The sample is then tilted by 52∘ relative to the Xe beam and
a narrower column beam is used to mill beneath the desired section of the
film, creating an inverted pyramid shape of YIG (Fig. 1(a)). Before the sample
is fully removed from the surface of the film, the finger of a nanomanipulator
approaches the material’s edge, where Pt is deposited, connecting the sample
to the finger. The pyramid-shaped YIG can then be extracted on the
nanomanipulator, as shown in Fig. 1(b). After the sample is fixed to the
nanomanipulator, it can be fully removed from the surface and moved to a
stage, where finer milling can be performed to achieve the desired shape (Fig.
1(c)). Afterwards, the sample is removed again by the nanomanipulator and
placed on top of prefabricated superconducting microwave devices (Fig. 1(d)).
Currents for Xe milling ranged from 1.8 to 180 nA. For the three samples
(#1-3) reported in this work, pieces of YIG were shaped into rough rectangular
cuboid shapes with dimensions ranging from 5 to 50 $\mu$m. The measured
dimensions for each sample are listed in Table 1. These pieces were placed,
with micron precision, on top of NbN microwave devices for spectral
measurement. The YIG samples were fixed to these devices using precise Pt
deposition. When necessary, ion milling was used to remove excess Pt that
could short the microwave devices (an example of this can be seen in Fig. 1(e)
as dark lines in the gaps between the central inductive lines and
interdigitated capacitors of the superconducting resonator devices).

Ferromagnetic resonance spectra were measured on YIG samples 1 and 2. The
samples were placed on top of NbN coplanar waveguide transmission lines with a
10 $\mu$m width, 6 $\mu$m gap geometry (see insets of Fig. 2(a) and (b)). The
devices were then placed inside Cu boxes for microwave measurement and mounted
inside a superconducting solenoid within an adiabatic demagnetization
refrigerator (see supplement Fig. S1(a) for a diagram of the measurement
setup). The solenoid field was calibrated using an electron spin resonance
measurement of a known materialVoesch  _et al._ (2015) (see supplement Fig.
S1(b) and text). Microwave transmission, $S_{21}$, was measured by a vector
network analyzer (VNA) sending signals to ports on the NbN transmission line.
The VNA signals were attenuated by 20 dB at 70 K, 4 K, and 0.5 K stages before
reaching the transmission line. At the 4 K stage, a high-electron-mobility
transistor (HEMT) provided 40 dB amplification to signals after passing
through the chip. An isolator placed before the HEMT prevented contamination
of transmission signals from spurious reflections back to the transmission
line. For some measurements, additional 45 dB amplification was provided by a
room temperature amplifier. VNA excitations ranged from -30 to 0 dBm, and no
power dependence was observed in this range for any samples. Ferromagnetic
resonance (FMR) was then measured at 3.2 K and 80 mK by applying a static
magnetic field parallel to the transmission line via the superconducting
solenoid. For all measurements presented, the signals are normalized with
respect to the microwave background signal.

As shown in Fig. 2(a-c), the FMR lines appear in the spectrum as absorbing
resonances whose frequency increases with applied field. The spectra have a
strong dependence on sample shape and size. For example, sample 1, with the
largest volume, exhibits several spin-wave modes (Fig. 2(a)). We associate
these additional resonances with a combination of magnetostatic surface spin
waves and forward and backward volume magnetostatic spin wavesSerga, Chumak,
and Hillebrands (2010). However, identification of the modes is difficult due
to the imperfect cuboid sample shape and potential effects on the spin waves
from the coplanar waveguide geometry of the superconducting transmission
lineGolovchanskiy  _et al._ (2019). On the other hand, if the width and
thickness (i.e. $x$ and $y$ dimensions) of the YIG are reduced, such as for
sample 2, then modes become more isolate, leaving only one dominant mode in
the spectrum (Fig. 2(b)). We associate this resonance in the spectrum of
sample 2 with the Kittel modeKittel (1948). Our main focus is on the Kittel
mode, whose resonance frequency is described according to the Kittel
equationKittel (1948)

$\omega_{FMR}=\gamma\mu_{0}\sqrt{\left[H_{eff}+(D_{x}-D_{z})M_{s}\right]\cdot\left[H_{eff}+(D_{y}-D_{z})M_{s}\right]}.$
(1)

Here $\gamma$ is the gyromagnetic ratio, $\mu_{0}$ is the permeability of free
space, $M_{s}$ is the saturation magnetization, and $D_{x}$, $D_{y}$, and
$D_{z}$ are demagnetization factors, which account for the effect of the
sample shape. $H_{eff}=H+H_{k}$ is the effective field consisting of the
external field $H$ produced by the superconducting solenoid applied along the
$z$ direction and the field $H_{k}$ associated with the YIG magnetocrystalline
anisotropy. The field-dependent resonance frequency for sample 2 can be fit to
this equation to determine the saturation magnetization of the PFIB-processed
YIG. First, however, the demagnetization factors are calculated for the cuboid
shape of the sample by using equations supplied in Ref. Aharoni, 1998. Using
these factors, which are listed in Table 1, the saturation magnetization is
determined to be $234\pm 4$ mT in reasonable agreement with other measurements
on YIG near zero temperature (see supplement Fig. S3(b) and Refs. Boventer
_et al._ , 2018; Mihalceanu  _et al._ , 2018). The magnetocrystalline
anisotropy, which depends on $M_{s}$, is determined to be $\mu_{0}H_{k}\approx
30$ mT and will be discussed in more detail below. However, due to the
weakness of the FMR signal, a determination of the coupling strength is not
possible, and therefore an extraction of the magnon linewidths from the FMR
data is unreliable. Nevertheless, from fits of the resonance signals of
samples 1-3, we can determine a linewidth range of 15-40 MHz. Furthermore, as
shown in Fig. 2(c), we find that the FMR modes show little to no change
between $2.9$ K and $80$ mK, indicating saturation of the linewidth in the
zero-temperature limit.

Figure 2: (a) Spectral measurement of sample 1 on top of a 10 $\mu$m wide NbN
transmission line at 3 K. Inset: SEM image of sample 1. The $x$ and $z$ axes
of the sample are labelled. The $y$ axis (unlabelled) is oriented
perpendicular to the NbN film surface. (b) Measurement of sample 2 on top of a
10 $\mu$m wide NbN transmission line at 2.9 K. Inset: SEM image of the sample
2 with labelled axes. (c) FMR resonances of sample 1 at $\mu_{0}H=160$ mT. The
FMR linewidths show no significant change from 2.9 K to 80 mK.

The main result of this work is the achievement of strong coupling between the
FMR modes of the PFIB processed YIG and the photonic mode of a superconducting
resonator. A scanning electron microscopy (SEM) image of the lumped-element
resonator devices, which consist of LC circuits comprised of interdigitated
capacitors connected by an inductive line, is shown in Fig. 1(e). Using the
PFIB processing techniques, YIG was placed on the center of the inductive line
of the shown resonators. At this position, the field profile derived from
microwave simulations, such as shown in Fig. 1(f), confirms that the YIG is
exposed to a uniform magnetic excitation field parallel with the chip surface
and directed across the line. This ensures the strongest coupling of both
resonant systems as well as that the FMR condition is met by having the
excitation field perpendicular to the externally applied field from the
solenoid, which is applied parallel to the inductive line of the resonator. Of
the devices shown in Fig. 1(e), we focus on the YIG-resonator device which
exhibited well-defined signatures of strong coupling; the spectra of other
select devices on the same chip are shown in Fig. S2 of the supplement. The
device of interest, which is called sample 3, has a zero-field resonance
frequency of $\omega_{p0}/2\pi=12.517$ GHz with a full-width at half-maximum
linewidth of $\Gamma/\pi=9.2$ MHz. At zero field, the loaded quality factor is
$1250$ after placement of the YIG. The resonance exhibits Fano-likeFano
(1961); Limonov  _et al._ (2017) behavior (see Fig. S3(a) in the supplement),
indicating the presence of interference. Such interference could be due to
imperfections created by the placement of YIG or coupling of the resonator to
stray resonances in the background transmission.

Spectral measurements of the device were performed using the same methods as
used to measure FMR of samples 1 and 2. As shown in Fig. 3, when field is
applied to the chip, the YIG FMR frequency increases, crossing the resonator
frequency. When the two resonances coincide, coupling lifts the degeneracy of
the system, resulting in an avoided crossing of modes in the spectrum. For our
device, we observe several avoided crossings corresponding to coupling between
the resonator photons and different spin-wave modes. The frequency separation
between the branches of the avoided crossings is related to the strength of
coupling, and this can be described generally by an equation of the formHuebl
_et al._ (2013); Macêdo  _et al._ (2021)

$\omega_{\pm}=\frac{1}{2}\left[\omega_{p}+\omega_{FMR}\pm\sqrt{(\omega_{p}-\omega_{FMR})^{2}+4g^{2}}\right].$
(2)

Here, $\omega_{p}$ is the resonator photon frequency and $g$ is the coupling
constant. Figure 3 shows fits of the branches of the largest avoided crossing
using Eq. 2. We use a least-squares fitting procedure to independently fit the
two branches. Saturation magnetization $M_{s}$ and $g$ are used as fit
parameters while $\omega_{p}/2\pi$ is fixed at 12.494 GHz. Although the
resonator frequency has a field dependence due to the proliferation of
vortices in the superconductor,Bothner  _et al._ (2012); Li  _et al._ (2019)
the field range for the avoided crossing is small enough that an adequate fit
can be accomplished without including this effect. From fits of the two
branches result, we find that $\mu_{0}M_{s}=205\pm 5$ mT and $g/2\pi=63\pm 5$
MHz for the largest avoided crossing, where the errors reflect the
disagreement between the two independent fits of each branch.

The coupling constant itself has been quoted as a function of the spin
density, $\rho$, taking the formHuebl  _et al._ (2013)

$g=\frac{\gamma}{2}\sqrt{\frac{\mu_{0}\hbar\omega_{p}\rho V_{m}}{2V_{p}}},$
(3)

where $V_{m}$ is the volume of the magnetic sample and $V_{p}$ is the
resonator’s mode volume. Here, we take the spin density to be $\rho=2\times
10^{22}$ cm-3 (appropriate for good-quality YIGHuebl  _et al._ (2013)) and
$V_{p}=1.59\times 10^{-5}$ cm3 (estimated using the finite-element simulations
package COMSOL Multiphysics®). Substituting these values into Eq. 3, we find
$g=70\pm 3$ MHz which is in reasonable agreement with the fits of the
experimental data. The coupling $g$ can also be theoretically derived using
electromagnetic perturbation theory (see Ref. Macêdo  _et al._ , 2021 and also
the supplement). In this case, $g$ is defined in terms of the magnetization
rather than spin density. Using $\mu_{0}M_{s}=205$ mT, the coupling is
predicted to be $g=68\pm 3$ MHz, in close agreement with the value based on an
assumed spin density.

Figure 3: Spectral measurement of the YIG-resonator device (sample 3) as a
function of external applied field $H$. Several avoided crossings occur as
each YIG FMR mode crosses the NbN resonator frequency. The black dashed lines
represent fits of the branches of the largest anticrossing based on Eq. 2. The
gap between the two branches corresponds to twice the effective coupling. For
the largest anticrossing, the effective coupling is estimated to be
$g/2\pi=63\pm 5$ MHz.

Without the inclusion of magnetocrystalline anisotropy, the saturation
magnetization of the PFIB-processed YIG shows consistent values of
$\mu_{0}M_{s}\sim 300$ mT for samples 2 and 3. Since the Kittel mode cannot be
reliably identified for sample 1, we cannot provide a proper estimate of
$M_{s}$. Nevertheless, in comparison to other studies on YIG in the sub-Kelvin
regimeBoventer  _et al._ (2018); Kosen  _et al._ (2019); Maier-Flaig  _et al._
(2017), the extracted values from our samples would initially seem much higher
than previously reported. While demagnetization factors are taken into
account, the high $M_{s}$ values imply that other effects such as
magnetocrystalline anisotropy should be considered. At 4.2 K, the first and
secondary cubic magnetocrystalline anisostropy constantsStancil and Prabhakar
(2009) for YIG are $K_{1}=-2480$ J/m3 and $K_{2}=-118$ J/m3. Although the
samples are cuboid in shape, the internal anisotropy fields can be
approximated by a relation for a (111)-oriented 2D filmRoss, Kostylev, and
Stamps (2011); Lee  _et al._ (2016). While the exact calculation of these
fields depends on the in-plane crystallographic orientation, which is lost
during the PFIB processing, an order of magnitude approximation is
$\mu_{0}H_{k}\approx-2K_{1}/M_{s}-2K_{2}/M_{s}$. By including this
$M_{s}$-dependent field term in Eq. 1, we determine $\mu_{0}H_{k}\approx 30$
mT and saturation magnetization values $\mu_{0}M_{s}=234\pm 4$ mT (sample 2)
and $205\pm 5$ mT (sample 3). These values of $M_{s}$ are more in line with
the saturation magnetization $\mu_{0}M_{s}=240\pm 1$ mT found for an
unstructured YIG film (see Fig. S3(b) in the supplement) and is also in the
range of previously reported values for pure YIG at $4$ K Mihalceanu  _et al._
(2018); Algra and Hansen (1982). Unfortunately, the PFIB-processed samples’
magnetic moments were below the sensitivity of the available SQUID
magnetometer. Previous studies indicate that ion implantation can
significantly decrease the magnetization of YIG Algra and Hansen (1982), so
without other means of investigation (e.g. Brillouin light scattering Agrawal
_et al._ (2013)), we cannot make a strong argument about presence of these
effects in our samples. However, a thorough study of anisotropies and
magnetization behaviors after PFIB fabrication is beyond the scope of this
article and is a subject for a future work.

The experimental and theoretical values of $g$ for sample 3 show relatively
good agreement with only $\sim 10\%$ difference. Since shape variation is
already accounted for in the uncertainty on the theoretical value, it cannot
completely account for the difference. Instead, the $10\%$ difference
indicates that either $V_{p}$ is slightly underestimated in the simulation or
the effective moment density $\rho$ is reduced due to damage from the PFIB
processing. Although neither $V_{p}$ nor $\rho$ can be measured directly,
there is some hint when comparing the spin-density and electromagnetic
perturbation theory based predictions of $g$. Since the electromagnetic
perturbation theory uses $M_{s}$, which is determined from fitting, the
general agreement between spin-density and electromagnetic perturbation
predictions of $g$ indicates that $\rho$ is close to the expected value for
ideal YIG, even though the determined values of $M_{s}$ are different for
samples 2 and 3. Therefore, it seems probable that surface damage from PFIB
processing does not strongly affect the effective spin density and that a
misestimation of $V_{p}$ is responsible for the disagreement between
experimental and theoretical determinations of $g$. Furthermore, this also
implies that the difference between $M_{s}$ for samples 2 and 3 may depend on
other factors such as $H_{k}$ and the crystalline orientation, which is
unaccounted for.

Based on the experimental determination of $g$ and assumed spin-density for
the first avoided crossing, the single-spin coupling constant can be
determined as $g_{0}/2\pi=g/2\pi\sqrt{N}=7.2$ Hz, where $N=\rho V_{m}$. In
addition, sample 3 exhibited at least two more at higher fields, indicating
the presence of strong magnon-photon coupling to higher order spin-wave modes.
Indeed, the cooperativityBoventer  _et al._ (2018) $C=g^{2}/\Gamma\kappa$ can
be calculated using the coupling $g$ and resonator and magnon linewidths
($\Gamma$ and $\kappa$, respectively) to confirm the strong coupling regime
for all three avoided crossings. While the respective linewidths can be
difficult to extract precisely from the data, they are estimated as
$\kappa=40\pm 5$ MHz and $\Gamma=6.4\pm 0.8$ MHz from fits of $S_{21}$ at
$\mu_{0}H=337.5$ mT (see supplement). Therefore the cooperativity is
approximately $C\approx 15$ for the first avoided crossing, confirming the
sample is firmly in the strong coupling regime where $C\gg 1$. Furthermore,
using the resonator linewidth information and following the method described
in the supplement, the average number of magnons $\langle n_{m}\rangle$ and
photons $\langle n_{p}\rangle$ in the system can be estimated as $\langle
n_{p}\rangle=\langle n_{m}\rangle=(6\pm 2)\times 10^{3}$.

While the first avoided crossing has a coupling $g/2\pi=63\pm 5$ MHz, each
subsequent avoided crossing shrinks in gap size and coupling with
$g/2\pi=18\pm 2$ MHz for the second crossing at $\mu_{0}H=362$ mT and
$g/2\pi=10\pm 10$ MHz for the third at $\mu_{0}H=380$ mT. Though the exact
nomenclature of these higher-order width or thickness spin-wave modes is
undetermined, in the context of Eq. 3, the smaller coupling rates imply the
effective $\rho$ is heavily reduced for these modes (i.e. fewer spins
participate in coupling) in comparison to the Kittel mode, in which the spins
precess together as a singular macrospin. The reduced coupling rates also
imply a steadily reducing cooperativity, and therefore it can be concluded
that future applications relying on higher order spin-wave modes will need to
achieve even larger coupling rates to remain in the strong coupling regime.
Nevertheless, the study here shows that such coupling rates may be possible
using PFIB-processed YIG.

Therefore, we have established a general method for creating on-chip hybrid
devices with high-quality spin-wave and superconducting components using PFIB
technology. Using these methods, we fabricated several devices and
demonstrated strong coupling between YIG FMR modes and a superconducting
resonator mode. This serves as a crucial step toward creating hybrid quantum
devices with spin-wave and superconducting components. The techniques used
here can be used to create any number of photon-magnon devices and circuits
such as hybrid logical gatesKostylev  _et al._ (2005), low-temperature
multiplexersVogt  _et al._ (2014), or spin-based quantum memory.

Supplementary Material

Further experimental, device characterization, and analysis details, along
with corresponding figures, can be found in the supplementary material.

Acknowledgements

Authors would like to thank I. I. Syvorotka for the fruitful discussions. This
work was supported by the European Research Council (ERC) under the Grant
Agreement 648011. R. Macêdo acknowledges support from the Leverhulme Trust,
and the University of Glasgow through LKAS funds. R. C. Holland was supported
by the Engineering and Physical Sciences Research Council (EPSRC) through the
Vacation Internships Scheme. S. McVitie and W. Smith acknowledge support from
EPSRC grant EP/M024423/1. RHH acknowledges support through ERC grant 648604
and a Royal Society Leverhulme Trust Senior Research Fellowship. The data that
support the findings of this study are available from the corresponding author
upon reasonable request.

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##  I Supplementary material for "Strong magnon-photon coupling with chip-
integrated YIG in the zero-temperature limit"

Below are supplemental figures for the main text of the manuscript "Strong
magnon-photon coupling with chip-integrated YIG in the zero-temperature
limit." Figures S1(a) and (b) show the diagram of the cryogenic and
measurement setups and calibration data for the magnet, respectively. Figures
S2(a) and (b) show the spectra of two other devices on the same chip as sample
3. Figure S3(a) shows the resonance at zero-field for the lumped-element
resonator of sample 3. Figure S3(b) shows a SQUID-based magnetization
measurement of an unstructured 100 $\mu$m thick film on a GGG substrate.
Figures S3(c) and (d) show sample transmission spectra $S_{21}$ for the first
avoided crossing of Fig. 3 of the main text. The transmission spectrum at
$\mu_{0}H=337.7$ mT is fit to determine the magnon linewidth $\kappa$ as
further explained below.

The experiments in the main text were carried out using a superconducting
solenoid to apply magnetic fields to the chips. The field strength was
calibrated using an electron spin resonance (ESR) measurement of
2,2-diphenyl-1-picrylhydrazyl (DPPH). This measurement was performed by
placing powdered DPPH on top of a superconducting CPW transmission line in a
similar fashion to a previous study on DPPH ESRVoesch  _et al._ (2015). The
ESR signal was measured over a range from 3.8 to 13.3 GHz and solenoid current
0 to 6 A. The frequency of the spin resonance is described as
$f_{DPPH}=\gamma_{e}\mu_{0}H$, where $\gamma_{e}=28$ GHz/T is the electron
gyromagnetic constant and $H$ is the solenoid field. The field is a function
of current ($I$) and turn denisity ($n$): $H=nI$. Therefore, a linear
dependence $f_{DPPH}=\gamma_{e}\mu_{0}nI$ is expected between frequency of the
DPPH ESR signal and the solenoid current. By fitting the frequency of the ESR
signal with respect to current, the calibration for magnetic field
$\mu_{0}H=mI+b$ can be extracted. Here, $m=\mu_{0}n$ is the calibration
constant, and $b$ is an offset field which can arise from residual currents
within the solenoid. For our solenoid magnet, we find $m=78.94\pm 0.02$ mT/A
and $b=2.32\pm 0.09$ mT. This is in relatively good agreement with the
expected calibration constant $m=83.3$ mT/A based on the design turn density
$n=66.3$ mm-1. For the calibration of the fields in the main text, we do not
include an offset field, since such contributions have a history dependence
and therefore change over timeGreen (1971). Nevertheless, we can estimate that
this field contribution only accounts for a $\sim 1\%$ error for the field
range of interest and does not strongly affect determinations of saturation
magnetization or change the conclusions drawn.

The sample dimensions and their relative errors are listed in Table 1. The
errors $\sigma_{i}$ ($i=x,y,z$) represent the deviations from an ideal cuboid
shape. In this case, they are calculated from differences on measurements of
opposite edges of each crystal. The demagnetization factors $D_{i}$ are
calculated from equations provided in Ref. Aharoni, 1998, and the errors
$\sigma_{D_{i}}$ on the demagnetization factors are then calculated as
$(\sigma_{D_{i}})^{2}=(\frac{dD_{i}}{dx}\sigma_{x})^{2}+(\frac{dD_{i}}{dy}\sigma_{y})^{2}+(\frac{dD_{i}}{dz}\sigma_{z})^{2}$.
Since the regression fits determining $M_{s}$, $g$, $\kappa$, and $\gamma$ do
not include errors on the demagnetization factors, the relevant errors are
calculated using standard error propagation techniques. For example, the error
$\sigma_{M_{s}}$ on $M_{s}$ is calculated as

$\left(\frac{\sigma_{M_{s}}}{M_{s}}\right)^{2}=\left(\frac{\sigma_{D_{x}}}{D_{x}}\right)^{2}+\left(\frac{\sigma_{D_{y}}}{D_{y}}\right)^{2}+\left(\frac{\sigma_{D_{z}}}{D_{z}}\right)^{2}+\left(\frac{\sigma_{H}}{H}\right)^{2}+\left(\frac{\sigma_{M_{fit}}}{M_{fit}}\right)^{2}.$
(S1)

Here, $M_{fit}=M_{s}$ is the value of the saturation magnetization from the
regression, and $\sigma_{M_{fit}}$ is the regression error. As stated above,
the field error $\sigma_{H}$ is approximately $1\%$ of the field $H$. Since
the demagnetization factors and field are determined independently, covariance
terms are not included for error propagation purposes. By this method, the
uncertainty values placed on given experimental results take into account
errors stemming from imperfect cuboid shapes.

As discussed in the main text, a new approach has been proposed for modeling
the magnon-photon coupling $g$ using electromagnetic perturbation theoryMacêdo
_et al._ (2021). In these terms, the coupling $g$ can be modeled as

$g=\sqrt{\frac{\chi\omega_{FMR}\omega_{p}V_{m}}{2V_{p}}},$ (S2)

where $\omega_{FMR}$ and $\omega_{p}$ are the ferromagnetic and photon
resonance frequencies, respectively, $V_{m}$ is the volume of the ferromagnet
(or ferrimagnet in the case of YIG), and $V_{p}$ is the mode volume of the
resonator. $\chi$ is the YIG susceptibility, which depends on the saturation
magnetization $M_{s}$ as $\chi=\frac{M_{s}}{(H+H_{k}+(D_{x}-D_{z})M_{s})}$
with magnetocrystalline anisotropy fields $H_{k}$ and demagnetization factors
$D_{x}$ and $D_{z}$. Using the $\mu_{0}M_{s}=205\pm 5$ mT and $\mu_{0}H=337.5$
mT for the condition that $\omega_{FMR}=\omega_{p}=12.494$ GHz, we calculate a
coupling constant $g=68\pm 3$ MHz. This value of $g$ is in close agreement
with the value $g=70\pm 3$ MHz, which is found using Eq. 3 of the main text
and an assumed spin density. Therefore, we can conclude that the effective
spin density $\rho$ of our YIG samples does not deviate strongly from the
ideal case of $\rho=2\times 10^{22}$ cm-3 for YIG.

To determine the magnon linewidth $\kappa$, the transmission spectrum $S_{21}$
is modeled asMorris  _et al._ (2017); Boventer  _et al._ (2018)

$S_{21}=1-\frac{2\Gamma_{c}}{i(\omega_{p}-\omega-\delta)+\Gamma+\frac{g^{2}}{i(\omega_{FMR}-\omega-\delta)+\kappa}}.$
(S3)

Here, $\Gamma$ is the total linewidth of the bare (i.e. YIG-free) resonator,
and $\Gamma_{c}$ is the coupling loss rate of the resonator to the
transmission line. Eq. S3 is adjusted to account for the notch-type geometry
of the resonator circuit. Figure S3(d) shows a fit of the transmission
spectrum at $\mu_{0}H=337.5$ mT within the first avoided crossing of Fig. 3.
Like the fits of $\omega_{\pm}$, the resonator frequency is set at the
predetermined value $\omega_{p}=12.494$ GHz. Since $\omega_{FMR}$ depends on
the saturation magnetization, $M_{s}$ is also set as a predetermined constant
$\mu_{0}M_{s}=205$ mT. Likewise, the coupling constant is fixed at $g=63$ MHz.
To accommodate for errors on $\omega_{p}$ and $M_{s}$, we include a free
parameter $\delta$ that allows for a shift of the horizontal frequency
position of the fit. Here, our main focus is the determination of the
linewidths, instead of precise determinations of $M_{s}$, $g$, or
$\omega_{p}$, which are instead determined from fits of $\omega_{\pm}$. The
errors on $\kappa$ and $\Gamma$ are determined by including additional error
propagation terms for $g$ and $M_{s}$ as explained above. From the fit in Fig.
S3(d), we determine $\Gamma_{c}=4.8\pm 0.6$ MHz, $\Gamma=6.4\pm 0.8$ MHz, and
$\kappa=40\pm 5$ MHz. The fit is shifted downward in frequency by
$\delta=35.7$ MHz.

From this determination of the linewidths, the average number of photons in
the resonator $\langle n_{p}\rangle$ and number of magnons $\langle
n_{m}\rangle$ in the coupled system can be calculated. For a bare resonator
without YIG, the number of photons in the resonator can be expressed
asSchneider (2020)

$\langle
n_{p}^{bare}\rangle=\frac{4P_{in}}{\hbar\omega_{p}^{2}}\frac{(\omega_{p}/\Gamma)^{2}}{(\omega_{p}/\Gamma_{c})}.$
(S4)

When the YIG FMR signal is tuned to match the resonator frequency, resonator
photons excite magnons in the system, and energy is split evenly between the
FMR and resonator systems. Therefore, by omitting higher order terms (e.g.
two-magnon processes) and assuming a one-to-one photon to magnon conversion,
particle numbers can be estimated as $\langle n_{p}\rangle=\langle
n_{m}\rangle=0.5\langle n_{p}^{bare}\rangle$. For the measurement of the
avoided crossing, the estimated power on the chip is -97 dBm and $\langle
n_{p}^{bare}\rangle=(1.2\pm 0.3)\times 10^{4}$. Therefore, the average number
of magnons and photons is $\langle n_{p}\rangle=\langle n_{m}\rangle=(6\pm
2)\times 10^{3}.$

Figure S1: (a) Diagram of the adiabatic demagnetization refrigerator
experimental setup for microwave transmission measurements. A VNA applied
microwave signals to the sample through a series of stainless steel (SS)
coaxial cables with 20 dB attenuators mounted on the 45 K, 4 K, and 0.5 K
temperature stages. The sample was mounted inside of a NbTi solenoid, oriented
such that the NbN transmission line is parallel to solenoid field. Before
returning to the VNA, signals pass through a circulator and 40 dB HEMT
amplifier. In some measurements an additional 45 dB gain room temperature
amplifier was also included. (b) Electron spin resonance measurement of
2,2-diphenyl-1-picrylhydrazyl (DPPH) as a function of solenoid current ($I$).
This measurement was used to calibrate the field of the superconducting
solenoid. The measurement was performed by placing DPPH on top of a NbN
transmission line similar to the one used for the YIG FMR measurements of the
main text. From the measurement of the DPPH electron spin resonance signal, we
find a field-current calibration of $\mu_{0}H=mI+b$ with $m=78.94\pm 0.02$
mT/A and an offset of $b=2.32\pm 0.09$ mT. Figure S2: Spectra for other
resonator devices on the same chip shown in Fig. 1(e) of the main text. These
resonances are at (a) 10.866 GHz and (b) 7.980 GHz and show a qualitatively
similar behavior as in Fig. 3. As magnetic field is applied, several avoided
crossings occur as various spin-wave modes cross the resonator frequency.
However, as the YIG sample dimensions are larger, many spin-wave modes (and
thus avoided crossings) are clustered together.  Figure S3: (a) Transmission
spectrum for a NbN lumped element LC resonator. The resonance is fit using a
Fano equation $S_{21}=A(q+\epsilon)^{2}/(1+\epsilon^{2})$, where
$\epsilon=(\omega-\omega_{0})/\Gamma$Fano (1961); Limonov  _et al._ (2017),
$A$ is the amplitude, $\omega_{0}$ is the resonance frequency, $\Gamma$ is the
linewidth, and $q$ produces the asymmetry of the Fano spectrum. The fit
provides a determination of the resonance frequency $\omega_{0}/2\pi=12.517$
GHz and full width at half maximum linewidth $\Gamma/2\pi=9.2$ MHz. (b)
Magnetization measurement of a 100 $\mu$m thick film of YIG grown on GGG
(black dots). Magnetization was measured at 4 K in a Physical Property
Measurement System using SQUID magnetometry. Saturation magnetization is
determined to be $\mu_{0}M_{s}=240\pm 1$ mT after subtracting away the
paramagnetic background (orange fits). (c) Transmission $S_{21}$ spectra at
different fields for the first avoided crossing shown in Fig. 3 of the main
text. The sample spectra shown are at fields before, within, and after the
first avoided crossing. (d) Fit of the $\mu_{0}H=337.5$ mT data within the
avoided crossing. Using Eq. S3, we fit the transmission to determine the
magnon and resonator linewidths $\kappa=40\pm 5$ MHz and $\Gamma=6.4\pm 0.8$
MHz, respectively.