# Enhancing the spin-photon coupling with a micromagnet

Xin-Lei Hei  Xing-Liang Dong  Jia-Qiang Chen  Cai-Peng Shen  Yi-Fan Qiao
Peng-Bo Li<EMAIL_ADDRESS>Shaanxi Province Key Laboratory of
Quantum Information and Quantum Optoelectronic Devices,
School of Physics, Xi’an Jiaotong University, Xi’an 710049, China

###### Abstract

Hybrid quantum systems involving solid-state spins and superconducting
microwave cavities play a crucial role in quantum science and technology, but
improving the spin-photon coupling at the single quantum level remains
challenging in such systems. Here, we propose a simple technique to strongly
couple a single solid-state spin to the microwave photons in a superconducting
coplanar waveguide (CPW) cavity via a magnetic microsphere. We show that,
strong coupling at the single spin level can be realized by virtual magnonic
excitations of a nearby micromagnet. The spin-photon coupling strength can be
enhanced up to typically four orders of magnitude larger than that without the
use of the micromagnet. This work can find applications in quantum information
processing with strongly coupled solid-state spin-photonic systems.

##  I  introduction

Hybrid quantum architectures based on degrees of freedom of completely
different nature have attracted much attention over recent years Xiang  _et
al._ (2013); Li  _et al._ (2015); Das  _et al._ (2017); Schütz  _et al._
(2020); Li  _et al._ (2020a); Bin  _et al._ (2020); Kubo  _et al._ (2011);
Faraon  _et al._ (2012); Bennett  _et al._ (2013); Li  _et al._ (2016); Li and
Nori (2018); Li  _et al._ (2020b); Blais  _et al._ (2020); Li  _et al._
(2021); Arcizet  _et al._ (2011); Li  _et al._ (2020c); Twamley and Barrett
(2010); Amsüss  _et al._ (2011); Grigoryan  _et al._ (2018); Julsgaard  _et
al._ (2013); Imamoğlu (2009); Kubo  _et al._ (2010), due to a variety of
applications in quantum technologies such as quantum networks Reiserer and
Rempe (2015), information processing Galindo and Martín-Delgado (2002); Blais
_et al._ (2020) and sensing Degen  _et al._ (2017). The interactions between
solid-state spins and microwave photons play a central role in hybrid quantum
systems Kubo  _et al._ (2010); Twamley and Barrett (2010); Amsüss  _et al._
(2011); Grigoryan  _et al._ (2018); Julsgaard  _et al._ (2013); Imamoğlu
(2009), which associate the solid-state spins as quantum memories Fuchs  _et
al._ (2011) and the photons as quantum information carriers Zhong  _et al._
(2020). Besides, they can be useful for fundamental investigation in quantum
mechanics, solid-state physics and quantum optics, which provide useful
platforms and tools to deepen the research of currently unexploited quantum
physics Mi  _et al._ (2018); Awschalom  _et al._ (2018).

To construct hybrid solid-state platforms, color centers in diamond are often
employed. For example, NV centers, one of the excellent color centers with
long coherence time and stable triplet ground states Aharonovich  _et al._
(2011); Lee  _et al._ (2017); Marcus  _et al._ (2013); Doherty  _et al._
(2014); Barry  _et al._ (2020); Bar-Gill  _et al._ (2013); Abobeih  _et al._
(2018), are frequently used for quantum storage Fuchs  _et al._ (2011) and
sensing Dolde  _et al._ (2011); Kolkowitz  _et al._ (2012). While the
realization of strong coupling is difficult due to the large mismatch between
the spatial extension of free-space photons and typical spins, the solid-state
spins can strongly couple to CPW resonators Roy  _et al._ (2017); Twamley and
Barrett (2010); Amsüss  _et al._ (2011), similar to cavity quantum
electrodynamics with atomsLiu  _et al._ (2014). However, current experiments
in hybrid systems often involve spin ensembles rather than single spins Verdú
_et al._ (2009); Kubo  _et al._ (2010); Amsüss  _et al._ (2011); Rabl  _et
al._ (2006), since the single spin-photon coupling strength is just around 10
Hz, which is far from the strong coupling regime Mi  _et al._ (2018); Blais
_et al._ (2020).

Here, we propose a feasible scheme to realize the strong spin-photon coupling
at the single quantum level via virtual excitations of magnons in a
micromagnet. Magnons, the energy quanta of spin waves, play an essential role
in quantum information processing and quantum sensing Zhang  _et al._ (2015);
Stancil and Prabhakar (2009); Zhang  _et al._ (2016a); Haigh  _et al._ (2016);
Lachance-Quirion  _et al._ (2019); Wolski  _et al._ (2020), with different
types of magnets such as sphere magnets Soykal and Flatté (2010); Tabuchi  _et
al._ (2014); Zhang  _et al._ (2014); Tabuchi  _et al._ (2015); Lambert  _et
al._ (2015, 2016); Bourhill  _et al._ (2016); Kostylev  _et al._ (2016); Wang
_et al._ (2018); Gonzalez-Ballestero  _et al._ (2020a); Gieseler  _et al._
(2020), film layer Sandweg  _et al._ (2011); Huebl  _et al._ (2013); Zhang
_et al._ (2016b); Li  _et al._ (2019a); Hou and Liu (2019); Wang  _et al._
(2020); Zhang  _et al._ (2020) and cylinder magnets Almulhem  _et al._ (2020).
Thanks to the small mode volume and high spin density of the Kittel mode in a
sphere micromagnet, it can focus the field energy to enable the strong spin-
magnon interaction Huillery  _et al._ (2020). Therefore, the Kittel mode in
the sphere magnet is frequently used in the magnon-cavity coupling system to
achieve the strong coupling between magnons and microwave cavity photons
Simons and Simons (2001); Wolf  _et al._ (2001); Verdú  _et al._ (2009);
Wesenberg  _et al._ (2009); Tabuchi  _et al._ (2016); Morris  _et al._ (2017);
Maier-Flaig  _et al._ (2017); Dany  _et al._ (2019); Li  _et al._ (2019b);
Martínez-Pérez and Zueco (2019). The nanoscale micromagnet, yttrium iron
garnet (YIG) sphere considered in this work, is well matched with the NV spins
and CPW cavities, and is very crucial to the strong interactions among the
three subsystems. We consider efficient coupling of magnonic excitations to a
single NV spin and a CPW cavity. We show that, even when the Kittel mode is
only virtually populated, the magnon-mediated interaction between the NV spin
and the microwave cavity mode can be modified significantly. The induced spin-
photon coupling strength can be enhanced up to typically four orders of
magnitude larger than that in the absence of the micromagnet. This regime may
provide a powerful tool for applications of quantum information processing
based on strong spin-photon interactions at the single quantum level.

##  II description of the system

###  II.1 The setup

As illustrated in Fig. 1(a), we consider a hybrid tripartite system where a
spherical micromagnet with radius $R$ is magnetically coupled to a NV center
and a CPW resonator simultaneously. Here, the magnetic microsphere is a YIG
sphere. The NV center is placed above the microsphere, which is close to the
surface of the CPW resonator.

Figure 1: (color online). (a) Schematic of the NV center (red circle with
arrow) coupled to a CPW cavity (blue) via a YIG sphere (gray). (b) Frequency
split of the NV center spin states and the resonant frequency of the Kittel
mode changed with the longitudinal magnetic field.

The magnetic microsphere supports spin waves, which always exist in the
ferrimagnetic or antiferromagnetic materials in the low energy limit. The spin
wave in a magnetic microsphere can be quantized by the dipolar, isotropic, and
magnetostatic approximations Gonzalez-Ballestero  _et al._ (2020a), with
magnons as the quanta. In our setup, only the Kittel mode in the YIG sphere is
considered, where all the spins in the magnetic sphere precess in phase and
with the same amplitude Tabuchi  _et al._ (2016). The Hamiltonian of the
Kittel mode (with the destruction operator $\hat{s}_{K}$) can be expressed as

$\displaystyle\hat{H}_{K}=\hbar\omega_{K}\hat{s}_{K}^{\dagger}\hat{s}_{K},$
(1)

Here, the frequency $\omega_{K}=|\gamma|B_{z}$ is controlled by the external
magnetic field, with the gyromagnetic ratio $\gamma=-1.76\times 10^{11}$
T-1s-1.

For the NV center, the spin-triplet $S=1$ ground states {$|0\rangle$, $|{\pm
1}\rangle$} are the eigenstates of the spin operator $\hat{S}_{z}$, with
$\hat{S}_{z}|{i}\rangle=i|{i}\rangle$ and $i=0,\pm 1$. In the presence of the
external magnetic field $\vec{B}_{z}$ oriented along the NV symmetry axis Maze
_et al._ (2011); Marcus  _et al._ (2013); Lee  _et al._ (2017); Li  _et al._
(2019c); Barry  _et al._ (2020), the degeneracy of the states $|{\pm
1}\rangle$ are split via the Zeeman effect. Taking $|{0}\rangle$ as the energy
reference, the Hamiltonian of the NV center can be expressed as

$\hat{H}_{NV}=\hbar\sum_{i=\pm 1}\omega_{i}|{i}\rangle\langle{i}|,$ (2)

with $\omega_{\pm 1}=D_{0}\pm|\gamma|B_{z}$ and the zero-field-splitting
$D_{0}=2\pi\times 2.87$ GHz. With proper magnetic fields, we can choose the
states $|{0}\rangle$ and $|{-1}\rangle$ as a spin qubit, which are selected to
be resonant with the magnon mode. The Hamiltonian of the NV center can be
simplified by

$\displaystyle\hat{H}_{NV}=\frac{\hbar}{2}\omega_{NV}\hat{\sigma}_{z},$ (3)

where we define the frequency as $\omega_{NV}\equiv\omega_{-1}$ and the spin
operator as
$\hat{\sigma_{z}}\equiv|{-1}\rangle\langle{-1}|-|{0}\rangle\langle{0}|$. The
state $|{+1}\rangle$ can be safely excluded due to its off-resonance with the
Kittel mode.

The CPW resonator is a one dimensional resonator whose frequency $\omega_{C}$
is related to the length of the transmission line segment. The electron
oscillating in the line cavity creates a variational electromagnetic field,
which can be quantized and interacts with other devices Wesenberg  _et al._
(2009); Bushev  _et al._ (2011); Jenkins  _et al._ (2013, 2014); Martínez-
Pérez and Zueco (2019). The corresponding Hamiltonian of photons in the CPW
resonator can be written as

$\displaystyle\hat{H}_{C}=\hbar\omega_{C}\hat{a}^{\dagger}\hat{a},$ (4)

where $\hat{a}$ is the annihilation operator of the photon mode. The photon
frequency is designed to match the frequencies of magnons and spins. That
means, in this setup, the frequencies of the three individual systems are
equivalent.

###  II.2 Interactions between spins and magnons

We now consider the quantization of the magnetic field induced by the magnetic
sphere. This magnetic field exists both inside and outside the micromagnet.
Here, we focus on the latter because of the location of spin qubit. From
classical electrodynamics, the magnetic field of a magnetic sphere with the
magnetization $\vec{M}$ can be expressed as

$\vec{B}_{m}(\vec{r})=\mu_{0}R^{3}/(3r^{3})\\{3(\vec{M}\cdot\vec{r})\vec{r}/r^{2}-\vec{M}\\},$
(5)

where $\mu_{0}=4\pi\times 10^{-7}$ T$\cdot$m$/$A is the vacuum permeability,
$R$ is the radius of the YIG sphere, and $\vec{r}=(r,\theta,\phi)$ is the
position vector relative to the centre of the magnetic sphere. After
quantizing the spin wave, the corresponding magnetization operator
$\hat{\vec{M}}$ of the Kittel mode is

$\displaystyle\hat{\vec{M}}={M_{K}}\left(\vec{\tilde{m}}_{K}\hat{s}_{K}+\vec{\tilde{m}}_{K}^{*}\hat{s}_{K}^{\dagger}\right),$
(6)

where $M_{K}=\sqrt{\hbar|\gamma|M_{s}/2V}$ is the zero-point magnetization,
$M_{s}$ is the saturation magnetization, and $V$ is the volume. Meanwhile, we
define the Kittel mode function as
$\vec{\tilde{m}}_{K}=\vec{e}_{x}+i\vec{e}_{y}$ with the unit coordinate vector
$\vec{e}_{x}$ and $\vec{e}_{y}$ (see more details in Appendix A). Remarkably,
there is no $z$ component in the mode function. From the above discussion, we
can get the quantized magnetic field $\hat{\vec{B}}_{m}$ as

$\displaystyle\hat{\vec{B}}_{m}(\vec{r})=$
$\displaystyle\frac{\mu_{0}R^{3}M_{K}}{3r^{3}}\Big{\\{}\big{[}(3C_{\theta}^{2}-1)\hat{X}+3S_{\theta}C_{\theta}\hat{P}\big{]}\vec{e}_{x}$
(7)
$\displaystyle+\big{[}3S_{\theta}C_{\theta}\hat{X}+(3S_{\theta}^{2}-1)\hat{P}\big{]}\vec{e}_{y}\Big{\\}},$

where we define $C_{\theta}=\cos\theta$, $S_{\theta}=\sin\theta$,
$\hat{X}=\hat{s}_{K}+\hat{s}_{K}^{\dagger}$ and
$\hat{P}=i(\hat{s}_{K}-\hat{s}_{K}^{\dagger})$ for convenience.

We then consider the interaction between the spin qubit and the Kittel mode in
the micromagnet. When a magnetic dipole is placed in a magnetic field, it
experiences a torque which tends to line it up parallel to the field. The
Hamiltonian associated with this torque can be naturally written as

$\hat{H}_{N-K}/\hbar=-g_{e}\mu_{B}\hat{\vec{B}}_{m}\cdot\hat{\vec{S}}$ (8)

with the electronic spin Landé factor $g_{e}$, the Bohr magneton $\mu_{B}$ and
the spin operator $\hat{\vec{S}}=(\hat{S}_{x},\hat{S}_{y},\hat{S}_{z})$. In
this work, we define the corresponding components of the spin operator as
$\hat{S}_{x}=\hbar(|{-1}\rangle\langle{0}|+|{0}\rangle\langle{-1}|)/2$,
$\hat{S}_{y}=\hbar(|{-1}\rangle\langle{0}|-|{0}\rangle\langle{-1}|)/2i$.
Assuming that the coupling strength is smaller than the resonance frequency,
the spin-magnon interaction under the rotation wave approximation can be
described by

$\displaystyle\hat{H}_{N-K}=-\hbar g(\hat{s}_{K}\hat{\sigma}^{+}+H.c.),$ (9)

where the spin operators are $\hat{\sigma}^{+}=|{-1}\rangle\langle{0}|$, and
$\hat{\sigma}^{-}=|{0}\rangle\langle{-1}|$. And the corresponding spin-magnon
coupling strength is expressed as

$\displaystyle
g=\sqrt{\frac{|\gamma|M_{s}}{12\pi}}\frac{g_{e}\mu_{0}\mu_{B}R^{3/2}}{(R+d)^{3}},$
(10)

where the distance between the spin qubit and the surface of the magnetic
sphere as $d=r-R$ (see Fig. 1(a)). The coupling strength is proportional to
the square root of the micromagnet volume with $d>R$, while the coupling
strength decreases slowly with $R>d$. In Fig. 2(a) and Fig. 2(b), we show the
dependence of the spin-magnon coupling as a function of $R$ and $d$.

Figure 2: (color online). (a)- (b) The couping strength $g/2\pi$ between the
individual NV center spin qubit and the Kittel mode in the YIG sphere versus
the distance $d$ and the radius $R$. (c) The coupling strength $\lambda/2\pi$
between the CPW resonator photon and the Kittel mode in the YIG sphere versus
the radius $R$.

The coupling rate keep reducing with the increase of $d$. While it increases
with the growth of $R$ utill the maximum. From the numerical result we find
that if we choose the YIG sphere with $R\sim 50$ nm and the distance $d\sim
10$ nm, then the coupling strength is about $\sim 1$ MHz.

###  II.3 Interactions between photons and magnons

We consider the quantizing form of the magnetic field generated by the CPW
resonator. For the single mode, the magnetic field operator is

$\hat{\vec{B}}_{c}=\vec{b}_{tr}(x,y)(\hat{a}+\hat{a}^{\dagger})/\sqrt{2}.$
(11)

Here, the mode function $\vec{b}_{tr}(x,y)$ varies strongly in space depending
on the CPW resonator geometry. To obtain the stronger interaction, the
magnetic sphere is in close proximity to the surface of the CPW resonator.
Then the mode function depends on the radius of the sphere
($\vec{b}_{tr}(R)$). From the discussion in Sec.II.1, the radius is about tens
of nanometers, which is less than the width of the CPW resonator ($\sim 1$
$\mu$m). Therefore, the variation of the mode function depending on the radius
can be ignored and $\vec{b}_{tr}(R)$ acts as an estimated constant ranging
from $35\sim 40$ $\mu$G.

We then expound the magnetic interaction between the CPW resonator and the
micromagnet. The classical exchange energy of a magnetic dipole with the
magnetic moment $\vec{m}$ in a uniform magnetic field $\vec{B}_{c}$ is
$U=-\vec{m}\cdot\vec{B}_{c}$. For a uniformly magnetized sphere, we treat it
as a magnetic dipole with $\vec{m}=V\vec{M}$. After replacing the classical
quantity with the magnetization operator $\hat{\vec{M}}$ and the magnetic
field operator $\hat{\vec{B}}_{c}$ in the expression above, the photon-magnon
interaction Hamiltonian is

$\hat{H}_{C-K}=-\hbar\lambda(\hat{s}_{K}+\hat{s}_{K}^{\dagger})(\hat{a}+\hat{a}^{\dagger}).$

Here, the homologous coupling strength is defined as

$\displaystyle\lambda=\sqrt{\frac{\pi|\gamma|M_{s}}{3\hbar}}b_{x}(R)R^{3/2}.$
(12)

We ignore the $y$ component of $\vec{b}_{tr}(R)$, since the magnetic field is
parallel to the $x$ axis. Then, the photon-magnon coupling strength $\lambda$
can be enhanced to around $1$ MHz with a proper geometrical size of the
magnetic microsphere (see Fig. 2(c)). Under the rotating wave approximation,
we can get the photon-magnon interaction Hamiltonian

$\displaystyle\hat{H}_{C-K}=-\hbar\lambda(\hat{s}_{K}^{\dagger}\hat{a}+H.c.).$
(13)

##  III realistic consideration and experimental parameters

From the above discussion, we can obtain the total Hamiltonian of the hybrid
spin-magnon-photon system, which can be expressed as

$\displaystyle\hat{H}=$
$\displaystyle\hbar\omega_{C}\hat{a}^{\dagger}\hat{a}+\hbar\omega_{K}\hat{s}_{K}^{\dagger}\hat{s}_{K}+\frac{1}{2}\hbar\omega_{NV}\hat{\sigma}_{z}$
(14) $\displaystyle-\hbar
g(\hat{s}_{K}\hat{\sigma}^{+}+H.c.)-\hbar\lambda(\hat{s}_{K}^{\dagger}\hat{a}+H.c.),$

where the first three terms correspond to the free Hamiltonian from Eq. (1),
Eq. (3) and Eq. (4), and the last two terms describe two interactions: one
between the Kittel mode and the NV center spin from Eq. (9), while the other
between the Kittel mode and the photon in the CPW resonator from Eq. (13). As
the Kittel mode is coupled both to the spin qubit and the CPW resonator
photon, it is possible to work as a quantum interface between the spin qubit
and the CPW resonator.

We now discuss the dynamics of the system in a realistic situation. In this
case, we take into account the dephasing of the NV center spin ($\gamma_{s}$),
the decays of the Kittel mode ($\gamma_{m}$) and the CPW resonator ($\kappa$).
As a result, the master equation of the total system can be expressed as

${\dot{\hat{\rho}}(t)}=-i/\hbar[{\hat{H},\hat{\rho}}]+\mathcal{L}[\hat{\rho}],$
(15)

where $\hat{\rho}$ is the density operator. The last term in above equation is
given by

$\displaystyle\mathcal{L}[\hat{\rho}]$ $\displaystyle=$
$\displaystyle{\gamma_{s}}\mathcal{D}[{\hat{\sigma}_{z}}]\hat{\rho}+\sum_{j}\\{(\bar{n}_{j}+1){\Gamma_{j}}\mathcal{D}[{{{\hat{o}}_{j}}}]\hat{\rho}$
(16)
$\displaystyle+\bar{n}_{j}{\Gamma_{j}}\mathcal{D}[\hat{o}_{j}^{\dagger}]\hat{\rho}\\}.$

For compactness we define
$\\{\Gamma_{m},\Gamma_{p}\\}\equiv\\{\gamma_{m},\kappa\\}$,
$\\{\hat{o}_{m},\hat{o}_{p}\\}\equiv\\{\hat{s}_{K},\hat{a}\\}$ and
$\mathcal{D}[\hat{o}]\hat{\rho}\equiv\hat{o}\hat{\rho}\hat{\rho}^{\dagger}-\\{\hat{o}^{\dagger}\hat{o},\hat{\rho}\\}/2$.
Here, we have introduced the thermal occupation number
$\bar{n}_{j}=(e^{\hbar\omega_{j}/kT}-1)^{-1}$ with frequencies
$\\{\omega_{m},\omega_{p}\\}=\\{\omega_{K},\omega_{C}\\}$ and environmental
temperature $T$ Li  _et al._ (2015); Gonzalez-Ballestero  _et al._ (2020b).
Considering the low temperature limit ($T\sim 10$ mT), the thermal occupation
number $\bar{n}_{j}$ is safely ingnored with the frequency $\omega_{j}\sim
1.4$ GHz. Then the dissipation of the Kittel mode and the photon mode
therefore can be simplified. Then, the master equation turns to

${\dot{\hat{\rho}}(t)}=-i/\hbar[{\hat{H},\hat{\rho}}]+{\gamma_{s}}\mathcal{D}[{\hat{\sigma}_{z}}]\hat{\rho}+\sum_{j}{\Gamma_{j}}\mathcal{D}[{{{\hat{o}}_{j}}}]\hat{\rho}.$
(17)

In the frame rotating at the individual spin qubit frequency $\omega_{NV}$,
the whole Hamiltonian is firstly transformed to the form as

$\displaystyle\hat{H}$ $\displaystyle=$
$\displaystyle\hbar\Delta_{1}\hat{s}_{K}^{\dagger}\hat{s}_{K}+\hbar\Delta_{2}\hat{a}^{\dagger}\hat{a}-\hbar
g\hat{s}_{K}\hat{\sigma}^{+}$
$\displaystyle-\hbar\lambda\hat{s}_{K}^{\dagger}\hat{a}+H.c.,$

where $\Delta_{1}=\omega_{K}-\omega_{NV}$ and
$\Delta_{2}=\omega_{C}-\omega_{NV}$ Liu  _et al._ (2014). In view of the large
detuning ($\Delta_{1}\gg g,$ $\lambda$), the Kittel mode can be eliminated Li
_et al._ (2012) and we can obtain the effective interaction between the NV
center spin qubit and the CPW resonator photon with the following Hamiltonian

$\displaystyle\hat{H}_{\mathrm{eff}}=$
$\displaystyle\hbar(\Delta_{2}-\beta^{2}\Delta_{1})\hat{a}^{\dagger}\hat{a}-\frac{1}{2}\hbar\alpha^{2}\Delta_{1}\hat{\sigma}_{z}$
(19) $\displaystyle-\hbar
g_{\mathrm{eff}}(\hat{a}^{\dagger}\hat{\sigma}^{-}+H.c.),$

where we difined the dimensionless parameters as $\alpha=g/\Delta_{1}$,
$\beta=\lambda/\Delta_{1}$ as well as the effective coupling srength
$g_{\mathrm{eff}}=g\lambda/\Delta_{1}$. This result shows that the interface
between the spin and the cavity photon is achieved via utilizing the Kittel
mode in the micromagnet as a medium.

We next discuss the effective coupled system. First, the dissipative
interaction system can be discribed with a reduced effective master equation
as

$\displaystyle\frac{{d{{\hat{\rho}}_{r}}(t)}}{{dt}}=$
$\displaystyle-\frac{i}{\hbar}[{{{\hat{H}}_{\mathrm{eff}}},{{\hat{\rho}}_{r}}}]+{\gamma_{s}}\mathcal{D}[{\hat{\sigma}_{z}}]{{\hat{\rho}}_{r}}$
(20)
$\displaystyle+{\gamma_{\mathrm{eff}}}\mathcal{D}[{{{\hat{\sigma}}_{-}}}]{{\hat{\rho}}_{r}}+{\kappa_{\mathrm{eff}}}\mathcal{D}[{\hat{a}}]{{\hat{\rho}}_{r}},$

where ${\gamma_{\mathrm{eff}}}=\ {\alpha^{2}}{\gamma_{m}}$ and
${\kappa_{\mathrm{eff}}}=\kappa+{\beta^{2}}{\gamma_{m}}$ are the effective
dacay rates, respectively. Here, the original dissipation rates are chosen as
$\gamma_{s}/2\pi\sim 1$ kHz Li  _et al._ (2019c), $\gamma_{m}/2\pi\sim 1$ MHz
Gonzalez-Ballestero  _et al._ (2020b) and $\kappa/2\pi\sim 6$ kHz Li  _et al._
(2015). And we take the dimensionless parameters as $\alpha\sim\beta\sim 0.1$
for the condition of eliminating the Kittel mode. We therefore estimate the
effective dacay rates as ${\gamma_{\mathrm{eff}}}\sim 10$ kHz and
${\kappa_{\mathrm{eff}}}\sim 16$ kHz. At the same time, the effective coupling
strength $g_{\mathrm{eff}}$ can be approximatively obtained as
$g_{\mathrm{eff}}\sim 0.1g$, which is larger than the effective dacay rates
${\gamma_{\mathrm{eff}}}$, ${\kappa_{\mathrm{eff}}}$ as well as the dephasing
of the NV center $\gamma_{s}$. Since the spin dephasing rate $\gamma_{s}$ is
much smaller than the effective decay rates ${\gamma_{\mathrm{eff}}}$ and
${\kappa_{\mathrm{eff}}}$, its effect on the dynamics of the system can be
neglected. Therefore, we can obtain the center result of this work: the
individual NV center spin qubit can strongly couple to the microwave photon in
a CPW resonator under the proper conditions. The spin-photon coupling is well
within the strong coupling regime.

Figure 3: (color online). The effective coupling strength
$g_{\mathrm{eff}}/2\pi$, the effective decay rates
${\gamma_{\mathrm{eff}}/2\pi}$ and ${\kappa_{\mathrm{eff}}}/2\pi$ depend on
the detuning $\Delta_{1}/2\pi$ with the value of (a) $R=50$ nm and (b) $R=100$
nm. And inversely these parameters depend on the radius of the micromagnet
sphere $R$ with the value of (c) $\Delta_{1}/2\pi=10$ MHz and (d)
$\Delta_{1}=20$ MHz. We choose the distance between the spin qubit and the
surface of the magnetic microsphere as $d=30$ nm in these figures.

Accurately, the effective coupling strength and decay rates depend on the
detuning $\Delta_{1}$ and the geometrical radius of the magnetic sphere $R$,
which is clearly shown in Fig. 3. The effective coupling rate
$g_{\mathrm{eff}}/2\pi$ and decay rate ${\kappa_{\mathrm{eff}}}/2\pi$ increase
with the increase of the detuning $\Delta_{1}/2\pi$ while they decrease with
the increase of $R$. Another effective decay rate
${\gamma_{\mathrm{eff}}/2\pi}$ decreases with the increase of
$\Delta_{1}/2\pi$ and $R$. There is always a large range where the effective
coupling strength exceeds both the effective decay rates with various values
of detuning $\Delta_{1}/2\pi$ and $R$. Thus, we can take the effective spin-
photon system into the strong coupling regime via tuning the relevant
parameters.

We proceed to analyze the effective spin-photon coupling from another point of
view. In order to compare the effective coupling strength with the effective
dacay rates, we employ three parameters
$g_{\mathrm{eff}}/\gamma_{\mathrm{eff}}$,
$g_{\mathrm{eff}}/\kappa_{\mathrm{eff}}$ as well as the cooperativity ${\cal
C}=g_{\mathrm{eff}}^{2}/\gamma_{\mathrm{eff}}\kappa_{\mathrm{eff}}$. The
patameter $g_{\mathrm{eff}}/\gamma_{\mathrm{eff}}$ increases with the increase
of the detuning $\Delta_{1}$ and the radius $R$ as shown in the Fig. 4(a). The
results show that the larger physical dimension of the magnetic microsphere
allows for stronger effective interaction between the spin qubit and the CPW
resonator photon, which makes experimental realization handy. There is a range
for the parameter $g_{\mathrm{eff}}/\kappa_{\mathrm{eff}}>1$ as shown in the
Fig. 4(e). The cooperativity $\cal C$ increases with the increase of the
detuning $\Delta_{1}$ and the radius $R$. And there is a very large range
where the cooperativity even exceeds 10 as shown in Fig. 4(d). Figs. 4(b) and
(c) respectively show that the three parameters are larger than 1 with
different detunings when the radius $R$ is within certain limits. Furthermore,
the results displayed in Fig. 3 and Fig. 4 are obtained when the NV spin qubit
is close to the magnetic microsphere ($d=30$ nm).

Figure 4: (color online). The contour maps show the parameters (a)
$g_{\mathrm{eff}}/\gamma_{\mathrm{eff}}$, (d) cooperativity and (e)
$g_{\mathrm{eff}}/\kappa_{\mathrm{eff}}$ as functions of the detuning
$\Delta_{1}/2\pi$ and the radius $R$. The more visualized relationship among
the three parameters, the detuning and the radius is shown in (b) and (c). And
the distance between the spin qubit and the surface of the magnetic
microsphere is chosen as $d=30$ nm as well. The short dots horizontal lines in
(b) and (c) point to the value 1.

We now explain the significance of the magnetic microsphere in this spin-
photon coupled system using the numerical simulation. The direct interaction
between the individual spin qubit and the single photon is very weak on
account of the low magnetic effect generated by a single microwave photon.
Nevertheless, the magnon mode is sensitive to weak magnetic fields. At the
same time, individual spin qubits can strongly couple to the magnon mode with
a small distance apart from the magnetic microsphere. Hence, the magnons play
a pivotal role in the realization of strong spin-photon coupling. The
effective interaction can be proportionally enhanced with the increase of the
volume of the magnetic microsphere within a certain range.

We numerically simulate the time evolution of the occupations of the spin
qubit, the Kittel magnon and the CPW resonator using the qutip package in
python Johansson  _et al._ (2013) as shown in the Fig. 5.

Figure 5: (color online). (a) The occupation map of the three parts in system
depends on the dimensionless time $gt/\pi$ with the parameters $\gamma_{m}\sim
g$, $\gamma_{s}\sim 0.1g$, $\kappa\sim 0.5g$, $g\sim\lambda$ and
$\Delta_{1}\sim 10g$. And the figure (b) shows the direct interaction between
the spin qubit and the CPW resonator photon without the magnon mode as an
intermediary.

The damped oscillations of the occupations of the spin and photon in Fig. 5(a)
show the strong coupling between them in the presence of magnons. While in the
absence of magnons, Fig. 5(b) shows an exponential decay curve without
oscillations. This result confirms the validity of the effective master
equation (20). Note that the population of the virtually excited Kittel mode
is nearly zero in the whole process. These results clearly show the
significant role played by the magnon in realizing the strong coupling between
the spin and the photon.

##  IV Conclusion

In conclusion, we have proposed a scheme for strongly coupling a single solid-
state spin like NV centers to the microwave photons in a CPW cavity via a
magnetic microsphere. We have shown that the strong coupling at the single
quantum level can be realized by virtual magnonic excitations of a nearby
micromagnet. In contrast to the case in the absence of the micromagnet, the
spin-photon coupling strength has been enhanced up to typically four orders of
magnitude. Here, the employment of magnons opens up intriguing perspectives
for magnonics and spintronics as well. This regime may facilitate much more
powerful applications in quantum information processing based on strongly
coupled solid-state spin-photonic systems.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
under Grants No. 92065105, No. 11774285 and the Natural Science Basic Research
Program of Shaanxi (Program No. 2020JC-02).

##  Appendix A Quantization of the spin wave

In this Appendix, we integrally show the quantization of the spin wave and
obtain the theoretical model of the magnon in a ferromagnetic microsphere.
First, in Sec. A.1, we introduce the original equation of motion for the
magnetization, the general Landau-Lifshitz equation, which is simplified under
several physical approximations. Then, we present the Walker mode in a ferrite
sphere in Sec. A.2. Based on the magnetostatic energy density expression, we
obtain the intrinsic Hamiltonian and quantization of the magnon modes in Sec.
A.3.

###  A.1 Spin wave equations in a magnetic sphere

First, we consider the spin waves with a continuous magnetization field
$\vec{M}(\vec{r},t)$ and the corresponding eletromagnetic field intensity
$\vec{E}(\vec{r},t)$ and $\vec{H}(\vec{r},t)$. The dynamics of spin waves
generally follows the Maxwell’s equations with the relationship between the
induced magnetization and the applied field. Now we start from the
phenomenological Landau-Lifshitz equation of motion for the magnetization
Stancil and Prabhakar (2009); Aharoni (2000)

$\frac{d}{dt}\vec{M}(\vec{r},t)=-|\gamma|\mu_{0}\vec{M}(\vec{r},t)\times\vec{H}_{\mathrm{eff}}(\vec{M},\vec{r},t),$
(21)

where the effective field $\vec{H}_{\mathrm{eff}}(\vec{M},\vec{r},t)$
comprises the Maxwellian field $\vec{H}(\vec{r},t)$ and the extra parts
$\vec{H}^{\prime}(\vec{M},\vec{r},t)$ as follows Stancil and Prabhakar (2009)

$\vec{H}^{\prime}(\vec{M},\vec{r},t)=\vec{H}_{\mathrm{ex}}(\vec{M},\vec{r},t)+\vec{H}_{\mathrm{an}}(\vec{M},\vec{r},t)+\vec{H}_{\mathrm{dm}}(\vec{M},\vec{r},t),$
(22)

Here, $\vec{H}_{\mathrm{ex}}$, $\vec{H}_{\mathrm{an}}$ and
$\vec{H}_{\mathrm{dm}}$ are effective fields due to exchange, anisotropy and
demagnetization induced by the magnetic dipole-dipole interactions,
respectively. The extra part of the effective field evidently depends on the
magnetization, which means that the Landau-Lifshitz equation is inhomogeneous.

We then assume that the magnet is in the saturated magnetic state along the
$z$ axis under the collinear field. Then the fluctuation of the magnetization
and the field is very small compared with the constant part Gonzalez-
Ballestero  _et al._ (2020a). Naturally, they are written as

$\displaystyle\vec{M}(\vec{r},t)$
$\displaystyle=M_{S}\vec{e}_{z}+\vec{m}(\vec{r},t),$ (23a)
$\displaystyle\vec{H}(\vec{r},t)$
$\displaystyle=H_{0}\vec{e}_{z}+\vec{h}(\vec{r},t)$ (23b)

Here, $\vec{m}\ll M_{S}$ and $\vec{h}\ll H_{0}$ are the dynamical variables to
be solved and $\vec{e}_{z}$ is the unit vector along the $z$ axis. Then we
consider the extra terms in the effective field. The exchange field is
associated with the domain wall interaction, which is less than the dipole-
dipole interaction when the micromagnet sizes is large compared with the
domain wall length. The magnetocrystalline anisotropy for a cubic material is
also negligible Aharoni (2000); Stancil and Prabhakar (2009); Maier-Flaig  _et
al._ (2017). As for the demagnetizing field, we assume
$\vec{H}_{\mathrm{dm}}=-(M_{S}/3)\vec{e}_{z}$, which is suitable for a
spherical magnet Aharoni (2000); Jackson (2007). The above expressions
conbined with the Eq. (21) lead to a more clear form of Landau–Lifshitz
equation as

$\displaystyle\frac{\dot{\vec{m}}(\vec{r},t)}{|\gamma|\mu_{0}M_{S}H_{0}}$
$\displaystyle-\vec{e}_{z}\times\left[\frac{\vec{m}(\vec{r},t)}{M_{S}}-\frac{\vec{h}(\vec{r},t)}{H_{0}}\right]+\left[\vec{e}_{z}+\frac{\vec{m}(\vec{r},t)}{M_{S}}\right]$
(24)
$\displaystyle\times\frac{(H_{0}-M_{S}/3)\vec{e}_{z}}{H_{0}}=\frac{\vec{h}(\vec{r},t)}{H_{0}}\times\frac{\vec{m}(\vec{r},t)}{M_{S}}$

Note that the variables $\vec{m}/M_{S}$ and $\vec{h}/H_{0}$ are small. Thus,
we can safely neglect the second order term in the above equation. Then the
Landau-Lifshitz equations is linearized as Stancil and Prabhakar (2009)

$\left[\begin{array}[]{c}\dot{m}_{x}(\vec{r},t)\\\
\dot{m}_{y}(\vec{r},t)\end{array}\right]=\left[\begin{array}[]{c}-\omega_{0}m_{y}(\vec{r},t)+\omega_{M}h_{y}(\vec{r},t)\\\
\omega_{0}m_{x}(\vec{r},t)-\omega_{M}h_{x}(\vec{r},t)\end{array}\right],$ (25)

where, two relevant system frequencies are

$\displaystyle\omega_{\mathrm{M}}$
$\displaystyle=|\gamma|\mu_{0}M_{\mathrm{S}},$ (26a) $\displaystyle\omega_{0}$
$\displaystyle=|\gamma|\mu_{0}(H_{0}-M_{S}/3).$ (26b)

We notice that the $z$ component vanishes because of the cross product with
vector $\vec{e}_{z}$. The result is not abnormal when we consider only small
fluctuations around the fully magnetized state which is along $z$ axis.

Finally, it comes to the magnetostatic approximation
$\nabla\times\vec{h}(\vec{r},t)\simeq 0$. In the Maxwell equations, the
electric field of the spin wave is uncoupled from $\vec{h}$ under this
circumstance. In addition, the approximation makes it easy to introduce the
magnetostatic potential through $\vec{h}(\vec{r},t)=-\nabla\psi(\vec{r},t)$.
Combined with the zero-divergence condition $\nabla\cdot\vec{b}=0$ in Maxwell
equations, the relation $\vec{b}=\mu_{0}(\vec{h}+\vec{m})$ allows one to
obtain the following equation for three acalar fileds Stancil and Prabhakar
(2009)

$\nabla^{2}\psi(\vec{r},t)=\partial_{x}m_{x}(\vec{r},t)+\partial_{y}m_{y}(\vec{r},t),$
(27)

which is the constraint inside the micromagnet. While the equation outside the
micromagnet is $\nabla^{2}\psi=0$. Therefore, the linear scalar equations Eq.
(25) and Eq. (27) completely describe the spin wave with the boundary
conditions, which is the continuity of the normal direction components of
$\vec{h}$ and $\vec{b}$, respectively. The spin-wave eigenmodes solved from
these equations are the magnetostatic dipolar spin waves or Walker modes
Aharoni (2000); Stancil and Prabhakar (2009).

###  A.2 Walker modes and Kittel modes

This section reveals the process of calculating the Walker modes. First, the
expressions of the magnetization and magnatic fields in terms of the
eigenmodes is shown by

$\displaystyle\vec{m}(\vec{r},t)=\sum_{\beta}\left[s_{\beta}\vec{m}_{\beta}(\vec{r})e^{-i\omega_{\beta}t}+\mathrm{c.c.}\right],$
(28a)
$\displaystyle\vec{h}(\vec{r},t)=\sum_{\beta}\left[s_{\beta}\vec{h}_{\beta}(\vec{r})e^{-i\omega_{\beta}t}+\mathrm{c.c.}\right].$
(28b)

Here, the eigenmode fields $\vec{m}_{\beta}(\vec{r})$ and
$\vec{h}_{\beta}(\vec{r})=-\nabla\psi_{\beta}(\vec{r})$ are characterized by a
series of mode indices $\\{\beta\\}$, an eigenfrequency $\omega_{\beta}$ and a
complex amplitude $s_{\beta}$ Fletcher and Bell (1959); Röschmann and Dötsch
(1977). Then the linearized Landau-Lifshitz equations turns to time-
independent forms

$\displaystyle i\omega
m_{x}(\vec{r})=\omega_{M}\partial_{y}\psi(\vec{r})+\omega_{0}m_{y}(\vec{r}),$
(29a) $\displaystyle i\omega
m_{y}(\vec{r})=-\omega_{M}\partial_{x}\psi(\vec{r})-\omega_{0}m_{x}(\vec{r}).$
(29b)

We can eliminate the scalar field $m_{x}(\vec{r})$ and $m_{y}(\vec{r})$
through these equations and Eq. (27). Then the formula only contains the
magnetostatic potential as

$\nabla^{2}\psi_{\text{out }}(\vec{r})=0,$ (30a)
$\left(1+\chi_{p}\right)\left(\frac{\partial^{2}}{\partial
x^{2}}+\frac{\partial^{2}}{\partial
y^{2}}\right)\psi_{\mathrm{in}}(\vec{r})+\frac{\partial^{2}}{\partial
z^{2}}\psi_{\mathrm{in}}(\vec{r})=0,$ (30b)

where $\psi_{\text{in }}$ and $\psi_{\text{out }}$ are the magnetostatic
potential inside and outside the micromagnet, respectively. Here, the diagonal
element of the Polder susceptibility tensor is defined as Stancil and
Prabhakar (2009)

$\chi_{p}(\omega)\equiv\frac{\omega_{M}\omega_{0}}{\omega_{0}^{2}-\omega^{2}}$
(31)

As for the outside situation, the general solution in the spherical
coordinates is given by

$\psi_{\text{out
}}(\vec{r})=\sum_{lm}\left[\frac{A_{lm}}{r^{l+1}}+B_{lm}r^{l}\right]Y_{l}^{m}(\theta,\phi).$
(32)

Here, the expansion coefficients $A_{lm}$, $B_{lm}$ are determined by the
boundary conditions. The spherical harmonics $Y_{l}^{m}(\theta,\phi)$
indicates the symmetry of the spin wave mode. While the potential inside the
sphere is more hard to solve. A set of nonorthogonal coordinates
$\\{\xi,\eta,\phi\\}$ is introduced to make it convenient. They fulfill

$\displaystyle x$
$\displaystyle=\sqrt{\chi_{p}}R\sqrt{\xi^{2}-1}\sin\eta\cos\phi,$ (33a)
$\displaystyle y$
$\displaystyle=\sqrt{\chi_{p}}R\sqrt{\xi^{2}-1}\sin\eta\sin\phi,$ (33b)
$\displaystyle z$
$\displaystyle=\sqrt{\frac{\chi_{p}}{1+\chi_{p}}}R\xi\cos\eta.$ (33c)

In the above coordinates, the solution of Eq. (30b) becomes available, which
is an expression including Legendre polynomials and spherical harmonics
Fletcher and Bell (1959); Röschmann and Dötsch (1977) as

$\psi_{\mathrm{in}}(\vec{r})=\sum_{lm}C_{lm}P_{l}^{m}(\xi)Y_{l}^{m}(\eta,\phi).$
(34)

In each term of the summation, the coefficients $C_{lm}$ depend on the
boundary conditions as well.

We then determine all the coefficients by the boundary conditions. First, the
term corresponding to $B_{lm}$ is not convergent at infinity, which should be
removed to make the potential $\psi$ regular. That indicates the first
condition, $B_{lm}=0$. Considering the potential on the surface of the sphere,
the coordinates are

$\xi\rightarrow\xi_{0}=\sqrt{\frac{1+\chi_{p}}{\chi_{p}}},\quad\\{\eta,\phi\\}\rightarrow\\{\theta,\phi\\}.$
(35)

Applying these coordinates to the two solution expressions Eq. (32) and Eq.
(34), we can use the boundary condition, the normal-direction-component
continuity of $\vec{h}$, and obtain the second condition

$A_{lm}=C_{lm}P_{l}^{m}\left(\xi_{0}\right)R^{l+1}.$ (36)

Similarly, we can apply the continuity of the normal component of the
$\vec{b}$ field to obtain the final condition Fletcher and Bell (1959)

$\left.\frac{\partial\psi_{\text{out }}}{\partial
r}\right|_{r=R}=\left.\frac{\xi_{0}}{R}\frac{\partial\psi_{\text{in
}}}{\partial\xi}\right|_{r=R}-\left.i\frac{\kappa_{p}}{R}\frac{\partial\psi_{\text{in
}}}{\partial\phi}\right|_{r=R}.$ (37)

Here, $\kappa_{p}(\omega)=\omega_{M}\omega/(\omega_{0}^{2}-\omega^{2})$ is the
off-diagonal element of the Polder susceptibility tensor Stancil and Prabhakar
(2009). Applying the above fomula and Eq. (36), the Walker mode eigenfrequency
fulfill the equation as Fletcher and Bell (1959); Röschmann and Dötsch (1977)

$\xi_{0}(\omega)\frac{P_{l}^{\prime
m}\left(\xi_{0}(\omega)\right)}{P_{l}^{m}\left(\xi_{0}(\omega)\right)}+m\kappa_{p}(\omega)+l+1=0.$
(38)

There are two results we can obtain from this equation: a) the eigenfrequency
$\omega$ is independent of the radius $R$; b) there are several solutions for
some pairs $\\{l,m\\}$ which are not positive and physical ($l=0$, for
instance). Generally, we can use the indices $\\{l,m\\}$ to mark the allowed
eigenmodes of the spin waves. The corresponding mode functions have been
explicitly calculated in Ref. Fletcher and Bell (1959).

###  A.3 The intrinsic Hamiltonian and quantization of the magnetostatic
dipolar magnon modes

We now show the quantization of the Walker modes from a phenomenological
micromagnetic energy functional Gonzalez-Ballestero  _et al._ (2020a)

$E_{m}(\\{\vec{m}\\},\\{\vec{h}\\})=\frac{\mu_{0}}{2}\int
dV\vec{m}(\vec{r},t)\cdot\left[\frac{H_{I}}{M_{S}}\vec{m}(\vec{r},t)-\vec{h}(\vec{r},t)\right]$
(39)

For convenience, we apply the linearized Landau-Lifshitz equations (25) to the
above expression and the energy only depends on $\vec{m}(\vec{r},t)$ in the
form

$\displaystyle E_{m}(\\{\vec{m}\\})=$
$\displaystyle\frac{1}{2|\gamma|M_{S}}\int
dV\left(m_{x}(\vec{r},t)\frac{\partial m_{y}(\vec{r},t)}{\partial t}\right.$
(40) $\displaystyle\left.-m_{y}(\vec{r},t)\frac{\partial
m_{x}(\vec{r},t)}{\partial t}\right).$

The key of quantization is to find the proper zero-point fluctuation and the
analogue energy expression with the harmonic oscillator. Thus, making use of
the magnetization expanded in terms of the eigenmodes, Eq. (28a), we transform
the above energy expression to Mills (2006); Walker (1957)

$E_{m}=\frac{1}{2\hbar|\gamma|M_{S}}\sum_{\beta}\hbar\omega_{\beta}\Lambda_{\beta}\left[s_{\beta}s_{\beta}^{*}+s_{\beta}^{*}s_{\beta}\right],$
(41)

where the coefficients have the following form

$\Lambda_{\beta}=2\mathrm{Im}\int dVm_{\beta y}(\vec{r})m_{\beta
x}^{*}(\vec{r}).$ (42)

Compared with the Hamiltonian of the harmonic oscillator, we can choose
adequate eigenmode normalization to fulfill
$\Lambda_{\beta}/(M_{S}|\gamma|\hbar)=1$ and the energy expression becomes
$E_{m}=\hbar/2\sum_{\beta}\omega_{\beta}[s_{\beta}s_{\beta}^{*}+s_{\beta}^{*}s_{\beta}]$.
After replacing the expansion coefficients with the bosonic magnon operators,
i.e., $s_{\beta}\rightarrow\hat{s}_{\beta}$ and
$s_{\beta}^{*}\rightarrow\hat{s}_{\beta}^{\dagger}$ with the commutation
relation $[\hat{s}_{\beta},\hat{s}_{\beta}^{\dagger}]=1$, we can obtain the
quantized magnon modes Hamiltonian
$\hat{H}_{m}=\sum_{\beta}\hbar\omega_{\beta}[\hat{s}_{\beta}^{\dagger}\hat{s}_{\beta}+1/2]$,
where the constant term is the analogue of the zero-point energy.

For simplicity, after defining a zero-point magnetization
$M_{0\beta}=\sqrt{\hbar|\gamma|M_{S}/\tilde{\Lambda}_{\beta}}$ and the
normalization constant $\tilde{\Lambda}_{\beta}=2\mathrm{Im}\int
dV\tilde{m}_{x}^{*}(\mathbf{r})\tilde{m}_{y}(\mathbf{r})$, the mode functions
are replaced as Gonzalez-Ballestero  _et al._ (2020a)

$\left[\begin{array}[]{c}\vec{m}_{\beta}(\vec{r})\\\
\vec{h}_{\beta}(\vec{r})\end{array}\right]\rightarrow{M}_{0\beta}\left[\begin{array}[]{c}\tilde{\vec{m}}_{\beta}(\vec{r})\\\
\tilde{\vec{h}}_{\beta}(\vec{r})\end{array}\right].$ (43)

The corresponding magnetization and magnetic field operators in the
Schrödinger picture are writen as

$\displaystyle\hat{\vec{m}}(\vec{r})=\sum_{\beta}{M}_{0\beta}\left[\tilde{\vec{m}}_{\beta}(\vec{r})\hat{s}_{\beta}+\mathrm{H.c.}\right],$
(44a)
$\displaystyle\hat{\vec{h}}(\vec{r})=\sum_{\beta}{M}_{0\beta}\left[\tilde{\vec{h}}_{\beta}(\vec{r})\hat{s}_{\beta}+\mathrm{H.c.}\right].$
(44b)

As for the Kittel mode, especially, the mode function is
$\tilde{m}_{K}=\vec{e}_{x}+i\vec{e}_{y}$ with the coordinate vector
$\vec{e}_{x}$ and $\vec{e}_{y}$. The zero-point magnetization is
$M_{K}=\sqrt{\hbar|\gamma|M_{S}/2V}$ with the saturation magnetization $M_{S}$
and the volume $V$ Gonzalez-Ballestero  _et al._ (2020a). Then the
corresponding magnetization operator is

$\hat{\vec{M}}={M_{K}}\left(\tilde{m}_{K}\hat{s}_{K}+\tilde{m}_{K}^{*}\hat{s}_{K}^{\dagger}\right).$
(45)

##  Appendix B Quantization of the magnetic field generated by a magnetic
sphere and the interaction Hamiltonian

In this appendix, we show the quantization of the magnetic field generated by
a magnetic sphere concretely.

We start from the classical electromagnetism result that the field of a
magnetic sphere with magnetization $\vec{M}$ can be described by

$\vec{B}(\vec{r})=\frac{\mu_{0}}{3}\frac{R^{3}}{r^{3}}\left[\frac{3(\vec{M}\cdot\vec{r})\vec{r}}{r^{2}}-\vec{M}\right],$
(46)

where $\vec{B}(\vec{r})$ is the magnetic field at
$\vec{r}=r\cos\theta\vec{e}_{x}+r\sin\theta\vec{e}_{y}$. We then introduce the
quantized magnetization operator from Eq. (45), i.e.,

$\hat{\vec{M}}=M_{K}\left[\left(\hat{s}_{K}+\hat{s}_{K}^{\dagger}\right)\vec{e}_{x}+i\left(\hat{s}_{K}-\hat{s}_{K}^{\dagger}\right)\vec{e}_{y}\right].$
(47)

Applying the above expression to Eq. (46), we can obtain the operator
expression of the magnetic field

$\displaystyle\hat{\vec{B}}_{m}(\vec{r})=$
$\displaystyle\frac{\mu_{0}R^{3}M_{K}}{3r^{3}}\Big{\\{}\big{[}(3C_{\theta}^{2}-1)\hat{X}+3S_{\theta}C_{\theta}\hat{P}\big{]}\vec{e}_{x}$
(48)
$\displaystyle+\big{[}3S_{\theta}C_{\theta}\hat{X}+(3S_{\theta}^{2}-1)\hat{P}\big{]}\vec{e}_{y}\Big{\\}},$

where we define $C_{\theta}=\cos\theta$, $S_{\theta}=\sin\theta$,
$\hat{X}=\hat{s}_{K}+\hat{s}_{K}^{\dagger}$ and
$\hat{P}=i(\hat{s}_{K}-\hat{s}_{K}^{\dagger})$ for convenience.

Then we consider the interaction between the spin and the quantized mgnetic
field described by
$\hat{H}_{N-K}=(-g_{e}\mu_{B}/\hbar)\hat{\vec{B}}\cdot\hat{\vec{S}}$ with the
spin operator
$\hat{S}=\hat{S}_{x}\vec{e}_{x}+\hat{S}_{y}\vec{e}_{y}+\hat{S}_{z}\vec{e}_{z}$,
which leads to the Hamiltonian

$\displaystyle\hat{H}_{N-K}=$ $\displaystyle-\hbar
g\big{[}(3C_{\theta}^{2}-1)(\hat{s}_{K}+\hat{s}_{K}^{\dagger})(\hat{\sigma}^{+}+\hat{\sigma}^{-})$
(49)
$\displaystyle+3iS_{\theta}C_{\theta}(\hat{s}_{K}-\hat{s}_{K}^{\dagger})(\hat{\sigma}^{+}+\hat{\sigma}^{-})$
$\displaystyle-3iS_{\theta}C_{\theta}(\hat{s}_{K}+\hat{s}_{K}^{\dagger})(\hat{\sigma}^{+}-\hat{\sigma}^{-})$
$\displaystyle+(3S_{\theta}^{2}-1)(\hat{s}_{K}-\hat{s}_{K}^{\dagger})(\hat{\sigma}^{+}-\hat{\sigma}^{-})\big{]},$

where the spin operators are defined as
$\hat{\sigma}^{\pm}=(\hat{S}_{x}\pm\hat{S}_{y})/\hbar$. Under the rotating
wave approximation, the imaginary terms in the second and third lines of the
above expression offset with each other, while the left parts can be
simplified to the final Hamiltonian $\hat{H}_{N-K}\simeq-\hbar
g(\hat{s}_{K}\hat{\sigma}^{+}+H.c.)$. Here, the coupling strength is defined
as

$\displaystyle
g=\sqrt{\frac{|\gamma|M_{s}}{12\pi}}\frac{g_{e}\mu_{0}\mu_{B}R^{3/2}}{(R+d)^{3}},$
(50)

where the distance between the spin qubit and the surface of the magnetic
sphere is $d=r-R$ (see Fig. 1(a)).

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