# Quantum parametric amplification of phonon-mediated magnon-spin interaction

Yan Wang1,2 Hui-Lai Zhang1,2 Jin-Lei Wu3 Jie Song4 Kun Yang1,2 Wei Qin5 Hui
Jing2,6∗ Le-Man Kuang2,6∗ 1College of Physics and Electronic Engineering,
Zhengzhou University of Light Industry, Zhengzhou 450001, China
2 Academy for Quantum Science and Technology, Zhengzhou University of Light
Industry, Zhengzhou 450001, China
3 School of Physics and Microelectronics, Zhengzhou University, Zhengzhou
450001, China
4 School of Physics, Harbin Institute of Technology, Harbin, 150001, China
5 Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering
Research, Wako-shi, Saitama 351-0198, Japan
6 Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of
Ministry of Education, Department of Physics and Synergetic Innovation Center
for Quantum Effects and Applications, Hunan Normal University, Changsha
410081, China
∗Corresponding authors (Hui Jing, email<EMAIL_ADDRESS>
Le-Man Kuang, email<EMAIL_ADDRESS>
###### Abstract

The recently developed hybrid magnonics provides new opportunities for
advances in both the study of magnetism and the development of quantum
information processing. However, engineering coherent quantum state transfer
between magnons and specific information carriers, in particular, mechanical
oscillators and solid-state spins, remains challenging due to the
intrinsically weak interactions and the frequency mismatch between different
components. Here, we show how to strongly couple the magnon modes in a
nanomagnet to the quantized mechanical motion (phonons) of a micromechanical
cantilever in a hybrid tripartite system. The coherent and enhanced magnon-
phonon coupling is engineered by introducing the quantum parametric
amplification of the mechanical motion. With experimentally feasible
parameters, we show that the mechanical parametric drive can be adjusted to
drive the system into the strong-coupling regime and even the ultrastrong-
coupling regime. Furthermore, we show the coherent state transfer between the
nanomagnet and a nitrogen-vacancy center in the dispersive-coupling regime,
with the magnon-spin interaction mediated by the virtually-excited squeezed
phonons. The amplified mechanical noise can hardly interrupt the coherent
dynamics of the system even for low mechanical quality factors, which removes
the requirement of applying additional engineered-reservoir techniques. Our
work opens up prospects for developing novel quantum transducers, quantum
memories and high-precision measurements.

mechanical parametric amplification, strong-coupling regime, hybrid quantum
system

PACS number(s): 42.50.Lc, 75.30.Ds, 63.20.Ls

1 Introduction

The studies on magnons, i.e., collective spin excitations of ordered magnets,
have recently made remarkable progress in exploring fundamental quantum
physics and realizing active quantum components for applications in quantum
science and technology Yuan  _et al._ (2022); Zare Rameshti  _et al._ (2022);
Barman  _et al._ (2021); Li  _et al._ (2020a); Kamra  _et al._ (2020); Harder
_et al._ (2021); Lachance-Quirion  _et al._ (2019). In this context, the
generation, manipulation and detection of magnons, i.e., the so called
magnonics, attracts increasing research interests for the perspective of
interfacing magnons with other quantum excitations. Examples include microwave
and optical photons Huebl  _et al._ (2013); Tabuchi  _et al._ (2014); Zhang
_et al._ (2014); Osada  _et al._ (2016); Zhang  _et al._ (2016a); Li  _et al._
(2019); Hou and Liu (2019); Xu  _et al._ (2020); Yuan  _et al._ (2020),
superconducting qubits Tabuchi  _et al._ (2015); Lachance-Quirion  _et al._
(2017, 2020), and acoustic phonons due to magnetostrictive interactions Zhang
_et al._ (2016b); Li  _et al._ (2018); Yu  _et al._ (2020); Shen  _et al._
(2022). The versatility of magnons opens new opportunities to further engineer
coherent interactions between magnons and other physical systems with
drastically different properties.

Hybrid quantum systems (HQSs) combine different physical components with
complementary functionalities, thus providing multitasking capabilities that
individual components cannot offer Kurizki  _et al._ (2015). In particular,
benefiting from the unprecedented tunability and compatibility, HQSs based on
magnonics, or say hybrid magnonics, promote the development of novel quantum
technologies, such as microwave-to-optical quantum transducers Hisatomi  _et
al._ (2016); Zhu  _et al._ (2020a); Chai  _et al._ (2022) and quantum
detection and sensing Cao and Yan (2019); Colombano  _et al._ (2020); Wolski
_et al._ (2020); Crescini  _et al._ (2020); Lu  _et al._ (2021). However, a
current challenge of functionalizing hybrid magnonics is how to coherently
couple the magnons to quantum systems that either weakly couple to magnons or
have large energy difference from magnon modes. Specifically, the former is
mainly caused by the inadequate spatial matching, thus making the coupling
strengths lower than intrinsic losses and decoherence of both systems. For the
latter, the coherent process of quantum state transfer can hardly take place
if the energy cannot be conserved, even in the presence of strong
interactions. A typical case is to coherently couple magnons to external
mechanical oscillators (corresponding to mechanical phonons aside from
acoustic phonons due to the deformation of the magnetic system itself). This
is generally difficult to achieve due to their dissimilar excitation energies
or frequency mismatch Zhang  _et al._ (2016b); Gonzalez-Ballestero  _et al._
(2020a, b). This cannot enable the coherent information transfer between
magnons and mechanical phonons, and thus may limit the development of
practical applications based on hybrid magnon-phonon systems Holanda  _et al._
(2018); Bozhko  _et al._ (2020), such as high-precision metrology of small
magnetic fields and mechanical displacements. In addition, recent studies have
found that magnons can strongly interact with solid-state spins such as
nitrogen-vacancy centers and silicon-vacancy centers in diamond Neuman  _et
al._ (2020); Wang  _et al._ (2021); Skogvoll  _et al._ (2021); Fukami  _et
al._ (2021); Xiong  _et al._ (2022). However, reaching the single magnon-spin
strong-coupling regime (SCR) often requires the spins to be positioned very
close to the magnet surface, which inevitably causes difficulties of
manipulating single spins.

In this work, we propose an experimentally feasible method for strongly
coupling magnons to mechanical phonons and to single solid-state spin in a
hybrid tripartite system. Our proposal is based on a hybrid device where a
micromechanical cantilever, pumped by a detuned parametric drive, interacts
simultaneously with a nanomagnet and a single nitrogen-vacancy (NV) center via
magnetic field gradients. We show that the magnon mode can be coupled, in a
coherent and tunable way, to the phonon mode by adjusting the mechanical
parametric drive (MPD). With experimentally accessible parameters, the magnon-
mechanical coupling can be exponentially enhanced, thus driving the initially
weak-coupling system into the SCR and even the ultrastrong-coupling regime
(UCR). The coupling enhancement can facilitate the extension of coherence time
of the quantum system where the mechanical mode acts as a long-lived quantum
memory. We also extend our protocol by coupling an NV spin to the mechanical
phonon. In the dispersive-coupling regime, we realize coherent quantum state
transfer between the nanomagnet and the NV center. Remarkably, the mechanical
noise amplified by the MPD can hardly corrupt the coherent and strong-coupling
dynamics of the proposed system without using additional engineered-reservoir
techniques that are generally required for suppressing the amplified noise Lü
_et al._ (2015); Qin  _et al._ (2019, 2018); Leroux  _et al._ (2018); Chen
_et al._ (2021); Tang  _et al._ (2022); Li  _et al._ (2020b); Wang  _et al._
(2022), which could simplify the experimental requirement and implementation.

Related forms of using the quantum parametric amplification (QPA) to
significantly enhance and controllably manipulate the interaction between
quantum objects have been studied in various coupled systems ranging from
optomechanics Lü  _et al._ (2015); Lemonde  _et al._ (2016); Qin  _et al._
(2019); Zhao  _et al._ (2020); Liu  _et al._ (2023), cavity or circuit QED Qin
_et al._ (2018); Leroux  _et al._ (2018); Wang  _et al._ (2019); Zhu  _et al._
(2020b); Groszkowski  _et al._ (2020); Wang  _et al._ (2020a, b); Qin  _et
al._ (2021); Chen  _et al._ (2021); Tang  _et al._ (2022); Villiers  _et al._
, trapped ions Ge  _et al._ (2019); Burd  _et al._ (2021); Affolter  _et al._
(2023) and hybrid spin-mechanical systems Li  _et al._ (2020b); Wang  _et al._
(2022); Pan  _et al._ (2023). Our work further extends and determines the
capabilities of exploiting the QPA to enhance the magnon-phonon and the
phonon-mediated magnon-spin interactions in a hybrid bipartite system. Notice
that a recent study has shown that the QPA can be used to enhance the direct
tripartite coupling among spins, magnons and phonons Hei  _et al._ (2023),
where markedly different physical model and coupling mechanism were considered
in a different hybrid setup. Our findings in achieving the effective
enhancement of the bipartite interaction between the magnon and an external
phonon provide a versatile tool for various intriguing applications, such as
the improvement of magnon-assisted ground-state cooling of the mechanical
motion Gonzalez-Ballestero  _et al._ (2020a); Asjad  _et al._ (2023), and the
preparation of highly entangled cat states between the magnon and the solid
spin in the weak-coupling regime Sharma  _et al._ (2021); Sun  _et al._
(2021). As for the enhancement of the bipartite interaction between the magnon
and the NV spin, our strategy strengthens the ability for processing
complicated quantum information tasks where both the NV spin and the magnons
can act as collaborate quantum memories. Our scheme may also find inspiring
applications to design quantum transducers, quantum memories and high-
precision measurements.

2 Quantum dynamics of hybrid magnon-phonon-spin system

Figure 1: (a) Schematic illustration of the proposed hybrid quantum system. A
micromechanical cantilever holding a spherical nanomagnet is parametrically
driven by a pump source that electrically modulates its spring constant. The
nanomagnet interacts with an external magnetic field, which has a homogeneous
component $\textbf{B}_{z}$ and a linear gradient $\textbf{B}_{0}(x)$. A magnet
tip attached underneath to the cantilever produces a strong magnetic gradient
$\textbf{B}_{1}(x)$ that couples the motion of cantilever to a lower single NV
center via the Zeeman effect. Dimensions of each components are not to scale
for visual clarity. (b) Schematic of the physical model. The effective
coupling between the spin qubit of frequency $\omega_{\mathrm{NV}}$ and the
magnon mode of frequency $\omega_{K}$ is mediated by the mechanical mode of
frequency $\omega_{x}$, on which a two-phonon (parametric) drive is applied to
amplify its zero-point fluctuations. This results in an exponential
enhancement of both the spin-mechanical coupling $\lambda$ and the magnon-
mechanical coupling $g$, thereby amplifying the resulting effective magnon-
spin interaction. (c) Finite-element simulations of the magnetic field lines
and the strength distribution generated by a Dy tip of 100 nm underside
diameter and height, and a YIG nanosphere of 100 nm radius, which are
separated by a distance of 300 nm along the $z$ direction. A diamond
cantilever with dimensions of $(\ell,w,t)=(4,0.1,0.02)$ $\mu$m is situated in
the middle. Only the $x$-$z$ plane of the spacial magnetic distributions is
illustrated here. Absolute values of the magnetic strength $|B_{1}(x)|$ (d)
and calculated magnetic gradient $b_{1}$ (e) along the $x$ direction for
$z=0$. Solid curves and hollow markers are results in the presence and absence
of the YIG sphere. A large magnetic gradient $b_{1}\approx 4.5\times 10^{7}$
T/m is obtained at the position of the NV spin (marked with a red dot in (c)).

We consider a hybrid tripartite system, as schematically illustrated in Fig.
1(a), where a spherical nanomagnet of radius $R$ is attached on top of a
singly-clamped micromechanical cantilever of dimensions ($\ell,w,t$) Vinante
_et al._ (2011); Burgess  _et al._ (2013); Vinante and Falferi (2013); den
Haan  _et al._ (2015), and a single NV center is placed under a magnetic tip
fixed at the bottom of the cantilever Kolkowitz  _et al._ (2012). The
nanomagnet interacts with an external magnetic field, which has a homogeneous
component $\textbf{B}_{z}=B_{z}\textbf{e}_{z}$, and a linear gradient
$\textbf{B}_{0}(x)=-b_{0}x\textbf{e}_{x}$, with $b_{0}$ the magnetic field
gradient. The homogeneous field $\textbf{B}_{z}$ is introduced to align the
magnetization of the nanomagnet to the $z$ direction, while the linear
gradient $\textbf{B}_{0}(x)$ is used to couple the nanomagnet to the
mechanical motion of the cantilever (see below). The motion of the magnetic
tip (assumed to be only along the $x$ direction) produces a strong magnetic
gradient $\textbf{B}_{1}(x)=-b_{1}x\textbf{e}_{x}$, which causes Zeeman shifts
of the NV center, giving rise to the magnetic coupling between the NV spin and
the mechanical motion of the cantilever (see below). Note here that the
magnetic gradient $\textbf{B}_{0}(x)$ has no effect on the spin-mechanical
coupling because it does not change the local magnetic field sensed by the
motionless NV spin. Additionally, we assume that the cantilever is
electrically pumped by a periodic drive that modulates the mechanical spring
constant in time Rugar and Grütter (1991); Szorkovszky  _et al._ (2011),
thereby amplifying the zero-point fluctuations of the mechanical motion. To be
specific, a time-varying voltage originating from a tunable oscillating pump
under the cantilever is applied to form a general capacitor between the two
electrodes, one of which is coated on the lower surface of the cantilever and
the other is placed on top of the pump, as illustrated in Fig. 1(a).

2.1 Free Hamiltonians for phonon, spin and magnon

We begin with the Hamiltonian of the mechanical cantilever. By modeling the
mechanical motion of the cantilever as a harmonic oscillator Rabl  _et al._
(2009, 2010) and following the standard quantization procedure Lemonde  _et
al._ (2016); Li  _et al._ (2020b); Wang  _et al._ (2022), the Hamiltonian of
the cantilever oscillator is obtained as

$\frac{\hat{H}_{\mathrm{mec}}}{\hbar}=\omega_{x}\hat{b}^{\dagger}\hat{b}-\Omega_{p}\cos(2\omega_{p}t)(\hat{b}^{2}+\hat{b}^{\dagger
2}),$ (1)

where $\hat{b}$ ($\hat{b}^{\dagger}$) is the annihilation (creation) operator
of the motional quanta, i.e., phonon, with frequency $\omega_{x}$, and
$\omega_{p}$, $\Omega_{p}$ are the pump frequency and amplitude, respectively.
Equation (1) shows that a two-phonon (i.e., parametric) drive of the
mechanical mode can be achieved by applying the time-dependent pump to the
mechanical oscillator. This detuned parametric drive, as detailed below,
modifies the eigenstates of the mechanical Hamiltonian to be squeezed Fock
states, whose amplified fluctuations in the anti-squeezed quadrature yield
larger interactions with both the magnon and the NV spin [see Fig. 1(b)]. In
addition, working with the typical mechanical zero-point fluctuations of
hundreds of femtometers, the proper distances between the two electrode
plates, e.g., tens of nanometers, and the accessible parameters with respect
to the electric drive module, the nonlinear amplitude $\Omega_{p}$ of up to
GHz could be realized Li  _et al._ (2020b).

For the single NV center, its electronic ground state is a $S=1$ spin triplet
with basis states $\left|m_{s}\right\rangle$, where $m_{s}=0,\pm 1$. The zero-
field splitting between the degenerate sublevels $\left|m_{s}=\pm
1\right\rangle$ and $\left|m_{s}=0\right\rangle$ is $D=2\pi\times 2.87$ GHz
Childress  _et al._ (2006); Kolkowitz  _et al._ (2012); Taylor  _et al._
(2008). We assume that the NV symmetry axis is along the $z$ direction. The
magnetic field $\textbf{B}_{z}$ lifts the degeneracy of the states $|m_{s}=\pm
1\rangle$, causing a Zeeman splitting $\delta=2g_{e}\mu_{B}B_{z}/\hbar$, where
$g_{e}\simeq 2$ is the Land$\acute{\mathrm{e}}$ factor and $\mu_{B}$ the Bohr
magneton. Taking the state $|0\rangle$ to be the energy reference, the free
Hamiltonian of the NV center has the form

$\frac{\hat{H}_{\mathrm{nv}}}{\hbar}=\sum_{j=\pm 1}\delta_{j}|j\rangle\langle
j|,$ (2)

where $\delta_{j}\equiv D\pm\delta/2$. By choosing a proper magnetic strength
$B_{z}$ such that
$\left|\delta_{+1}-\omega_{x}\right|\gg\left|\delta_{-1}-\omega_{x}\right|$,
the contribution $\propto$ $\left(\left|+1\rangle\langle
0\right|+\left|0\rangle\langle+1\right|\right)(\hat{b}^{\dagger}+\hat{b})$
oscillates rapidly and thus can be excluded from the dynamics. The Hamiltonian
(2) can then be rewritten as

$\frac{\hat{H}_{\mathrm{NV}}}{\hbar}=\frac{1}{2}\omega_{\mathrm{NV}}\hat{\sigma}_{z},$
(3)

where $\omega_{\mathrm{NV}}\equiv\delta_{-1}$ and
$\hat{\sigma}_{z}=|-1\rangle\langle-1|-|0\rangle\langle 0|$.

Next, we turn our attention to the spherical nanomagnet that supports the
magnetization-wave or spin-wave (magnon) modes Zare Rameshti  _et al._ (2022).
Under the dipolar, isotropic, and magnetostatic approximations Gonzalez-
Ballestero  _et al._ (2020b), one can obtain the magnetization eigenmodes,
known as the magnetostatic dipolar spin waves, or Walker modes for spherical
magnets Walker (1957, 1958); Röschmann and Dötsch (1977). The eigenfrequencies
of the Walker modes, $\omega_{\beta}\equiv\omega_{lmn}$, depend on the
external homogeneous field $B_{z}\equiv\mu_{0}H_{z}$, where $l=1,2,3,\dots$,
$m\in[-l,l]$, $n=0,1,2,\dots,n_{\mathrm{max}}(l)$, and $\mu_{0}$ is the vacuum
permeability. These Walker modes can be further quantized Mills (2006),
resulting in the free magnon Hamiltonian

$\frac{\hat{H}_{\mathrm{m}}}{\hbar}=\sum_{\beta}\omega_{\beta}\hat{s}_{\beta}^{\dagger}\hat{s}_{\beta},$
(4)

with the bosonic ladder operators
$[\hat{s}_{\beta},\hat{s}_{\beta^{\prime}}^{\dagger}]=\delta_{\beta\beta^{\prime}}$.
Correspondingly, the spin-wave magnetization operator in the
Schr$\mathrm{\ddot{o}}$dinger picture is given by

$\hat{\mathbf{m}}(\mathbf{r})=\sum_{\beta}\mathcal{M}_{0\beta}[\tilde{\mathbf{m}}_{\beta}(\mathbf{r})\hat{s}_{\beta}+\text{H.c.}],$
(5)

where $\tilde{\mathbf{m}}_{\beta}(\mathbf{r})$ is the mode function of the
spin wave, and
$\mathcal{M}_{0\beta}\equiv\sqrt{\hbar|\gamma|M_{S}/\tilde{\Lambda}_{\beta}}$
is the zero-point magnetization, with $|\gamma|=g_{e}\mu_{B}/\hbar$ the
absolute value of the gyromagnetic ratio, $M_{S}$ the saturation magnetization
and $\tilde{\Lambda}_{\beta}\equiv 2\operatorname{Im}\int
dV\tilde{m}_{x}^{*}(\mathbf{r})\tilde{m}_{y}(\mathbf{r})$ the normalization
constant.

In the present work, we focus only on the simplest Walker mode, i.e., the
spatially uniform Kittel mode Kittel (1948), $\beta\equiv\\{110\\}\equiv K$ .
In this case, the free Hamiltonian of the Kittel magnon mode is given by

$\hat{H}_{\mathrm{mag}}/\hbar=\omega_{K}\hat{s}^{\dagger}\hat{s},$ (6)

where $\omega_{K}=|\gamma|B_{z}$ is the eigenfrequency of the Kittel mode, and
the mode index $K$ has been omitted for simplicity. Considering that the
magnetized mode function of the Kittel mode takes a homogeneous form,
$\tilde{\mathbf{m}}_{\mathrm{K}}(\mathbf{r})=\mathbf{e}_{x}+i\mathbf{e}_{y}$,
the zero-point magnetization becomes
$\mathcal{M}_{K}=\sqrt{\hbar|\gamma|M_{S}/2V}$, with $V$ the nanomagnet
volume. We remark that there is no $z$ component in the magnetized mode
function. The corresponding magnetization operator for the Kittel mode can
then be obtained as

$\hat{\mathbf{m}}_{K}(\mathbf{r})=\mathcal{M}_{K}[(\mathbf{e}_{x}+i\mathbf{e}_{y})\hat{s}+\text{H.c.}].$
(7)

2.2 Interaction between magnon and phonon or spin and phonon

In order to couple the magnon mode of the nanomagnet to the mechanical motion
of the cantilever, we apply a spatially inhomogeneous magnetic field
$\textbf{B}_{0}(x)$. For a general driving magnetic field
$\mathbf{H}_{d}\left(\mathbf{r},t\right)$, it can increase the total energy of
the nanomagnet by Gonzalez-Ballestero  _et al._ (2020b)

$\Delta
E(t)=-\frac{\mu_{0}}{2}\int\text{d}V\left[M_{S}\textbf{e}_{z}+\mathbf{m}(\mathbf{r},t)\right]\cdot\mathbf{H}_{d}\left(\mathbf{r},t\right).$
(8)

Here, the first and second terms describe the interactions of the driving
field with the static magnetization $M_{S}\textbf{e}_{z}$ and the
magnetization of the spin waves $\mathbf{m}(\mathbf{r},t)$, respectively. To
obtain the magnon-mechanical interaction Hamiltonian of our system in the
Schr$\mathrm{\ddot{o}}$dinger picture, we substitute the dynamical variable
$\mathbf{m}(\mathbf{r},t)$ in Eq. (8) by its quantum operator
$\hat{\mathbf{m}}_{K}(\mathbf{r})$ given in Eq. (7), the variable
$\mathbf{H}_{d}\left(\mathbf{r},t\right)$ by
$\mathbf{H}_{0}(x)\equiv\mathbf{B}_{0}(x)/\mu_{0}$, and the variation in
magnetic energy $\Delta E(t)$ by operator $\hat{V}$. The resulting interaction
Hamiltonian is given by

$\hat{V}=\frac{b_{0}V\mathcal{M}_{K}}{2}(\hat{s}^{\dagger}+\hat{s})\hat{x},$
(9)

which describes the coupling between the Kittel magnon and the mechanical
motion along the $x$ direction. By further rewriting the position operator of
the mechanical oscillator as
$\hat{x}=x_{\mathrm{zpf}}(\hat{b}^{\dagger}+\hat{b})$, where
$x_{\mathrm{zpf}}=\sqrt{\hbar/2M_{\mathrm{eff}}\omega_{x}}$ is the zero-point
fluctuation amplitude, with $M_{\mathrm{eff}}=M+M_{m}+M_{t}$ covering the
effective masses of the cantilever ($M$), the nanomagnet ($M_{m}$) and the
magnetic tip ($M_{t}$), the magnon-mechanical interaction Hamiltonian becomes

$\hat{H}_{\mathrm{m}\text{-}\mathrm{m}}/\hbar=g(\hat{s}^{\dagger}+\hat{s})(\hat{b}^{\dagger}+\hat{b}),$
(10)

where $g\equiv b_{0}x_{\mathrm{zpf}}\mathcal{M}_{K}V/2\hbar$ is the magnon-
mechanical coupling rate.

For the proposed hybrid system in Fig. 1(a), we desire to provide a mechanism
where the spin-phonon coupling is determined only by the magnetic gradient
$\textbf{B}_{1}(x)$ originating from the vibration of magnetic tip, while the
inhomogeneous external field $\textbf{B}_{0}(x)$ is responsible only for the
magnon-phonon coupling. To this end, we first select a magnetic tip with a
size of $\sim$100 nm, and the magnetic substance is selected as Dysprosium
(Dy) due to a high saturation magnetization of up to 3.7 T Mamin  _et al._
(2012), which is larger than that of yttrium iron garnet (YIG) with
$\mu_{0}M_{S}\approx 0.74$ T. This allows us to obtain a strong magnetic
gradient exceeding $10^{7}$ T/m at short distances for the NV center, as
demonstrated in the simulation below. Moreover, we consider a specific
arrangement of magnets such that the magnetic tip is positioned, along the $z$
axis, at a distance of hundreds of nanometers away from the nanomagnet. This
can avoid the interplay of the magnetic fields generated by the magnetic tip
and the nanomagnet due to an adequate spatial separation. To verify this, we
numerically simulate the relevant magnetic fields and compute the value of
magnetic gradient using the finite-element analysis method, and the results
are shown in Figs. 1(c)-1(e). Figure 1(c) is an example of the simulated
magnetic field lines and strength distribution for a Dy tip with underside
diameter and height of 100 nm, and a YIG sphere with radius of 100 nm, which
are separated by 300 nm in the direction of $z$-axis. One can see in this
configuration that the field distribution around the Dy tip can hardly be
affected by the YIG sphere and vice versa. Moreover, Figs. 1(d) and 1(e) show
values of the magnetic strength $|B_{1}(x)|$ and the gradient $b_{1}$ along
the $x$ direction for $z=0$. Here, the effect of the YIG sphere on the Dy tip
is evaluated by comparing the values of $B_{1}(x)$ and $b_{1}$ after removing
the YIG sphere (labeled with “Air” in the plots). We find that both the
magnetic strength and the magnetic gradient remain unchanged in the presence
or absence of the YIG sphere, revealing the negligible influence of the YIG
sphere on the magnetic tip in the present configuration.

The oscillations of the mechanical cantilever as well as the Dy tip produce a
strong magnetic gradient that causes the Zeeman shifts of the NV center Li
_et al._ (2016); Sánchez Muñoz  _et al._ (2018); Zhou  _et al._ (2022),
thereby coupling the spin and mechanical degrees of freedom. The magnetic
interaction Hamiltonian can be written as

$\hat{H}_{\mathrm{int}}=\frac{\mu_{B}g_{e}}{\hbar}\hat{\textbf{S}}\cdot\textbf{B}(\textbf{r}_{0}),$
(11)

where $\hat{\textbf{S}}$ denotes the spin operator and $\textbf{r}_{0}$ the
position of the NV center. In our setup, the bending motion of the cantilever
(along the $x$ direction) modulates the local magnetic field sensed by the NV
center, giving rise to the spin-mechanical coupling with the interaction
Hamiltonian

$\hat{H}_{\mathrm{s}\text{-}\mathrm{m}}=-\frac{\mu_{B}g_{e}b_{1}}{\hbar}\hat{S}_{x}\hat{x}.$
(12)

After using again $\hat{x}=x_{\mathrm{zpf}}(\hat{b}^{\dagger}+\hat{b})$, this
interaction Hamiltonian becomes

$\hat{H}_{\mathrm{s}\text{-}\mathrm{m}}/\hbar=\lambda(\hat{\sigma}_{+}+\hat{\sigma}_{-})(\hat{b}^{\dagger}+\hat{b}),$
(13)

where $\lambda\equiv-\mu_{B}g_{e}b_{1}x_{\mathrm{zpf}}/\hbar$ denotes the
spin-mechanical coupling rate and the spin operators
$\hat{\sigma}_{+}=\hat{\sigma}_{-}^{\dagger}=|-1\rangle\langle 0|$. This spin-
mechanical coupling depends on the magnetic gradient $b_{1}$, which can be
adjusted by positioning the NV spin at the suitable distances from the Dy tip.
For instance, a large gradient $b_{1}\approx 4.5\times 10^{7}$ T/m can be
obtained for a distance of about 10 nm between the Dy tip and the NV spin, as
shown in Fig. 1(e).

To sum up, we obtain the total Hamiltonian for the proposed hybrid system

$\begin{split}\frac{\hat{H}_{\mathrm{hybrid}}}{\hbar}=&\hat{H}_{\mathrm{mec}}+\hat{H}_{\mathrm{NV}}+\hat{H}_{\mathrm{mag}}+\hat{H}_{\mathrm{m}\text{-}\mathrm{m}}+\hat{H}_{\mathrm{s}\text{-}\mathrm{m}}\\\
=&\omega_{x}\hat{b}^{\dagger}\hat{b}+\frac{1}{2}\omega_{\mathrm{NV}}\hat{\sigma}_{z}+\omega_{K}\hat{s}^{\dagger}\hat{s}\\\
&+g(\hat{s}^{\dagger}+\hat{s})(\hat{b}^{\dagger}+\hat{b})+\lambda(\hat{\sigma}_{+}+\hat{\sigma}_{-})(\hat{b}^{\dagger}+\hat{b})\\\
&-\Omega_{p}\cos(2\omega_{p}t)(\hat{b}^{2}+\hat{b}^{\dagger 2}).\end{split}$
(14)

3 Giant enhancement of phonon-mediated magnon-spin interaction

In this section, we show how to engineer coherent and enhanced magnon-
mechanical interaction and the consequent magnon-spin interaction mediated by
a squeezed phonon. Our approach is based on the simultaneous enhancement of
the magnon-mechanical and spin-mechanical couplings with the squeezed phonons
being virtually excited under some proper parameters. Since the direct
enhancement of the spin-mechanical coupling via the parametric amplification
of the mechanical motion has been discussed in detail elsewhere Li  _et al._
(2020b); Wang  _et al._ (2022), in the following we first show our new results
of extending the previous scheme Li  _et al._ (2020b); Wang  _et al._ (2022)
to involve the magnonic degree of freedom, and on this basis we derive the
enhanced magnon-spin coupling in the dispersive regime.

3.1 Exponentially enhanced magnon-phonon coupling in the squeezed frame

We first consider the coupling between the nanomagnet and the mechanical
oscillator by excluding the spin degree of freedom. In this case, the magnon-
mechanical Hamiltonian, in the frame rotating at $\omega_{p}$ and after
dropping the high-frequency oscillation terms, is given by

$\begin{split}\frac{\hat{H}_{\mathrm{mm}}}{\hbar}=&\Delta_{x}\hat{b}^{\dagger}\hat{b}+\Delta_{K}\hat{s}^{\dagger}\hat{s}-\frac{\Omega_{p}}{2}(\hat{b}^{2}+\hat{b}^{\dagger
2})\\\
&+g(\hat{b}\hat{s}^{\dagger}+\hat{b}^{\dagger}\hat{s}^{\dagger}e^{\mathrm{i}2\omega_{p}t}+\mathrm{H.c.}),\end{split}$
(15)

where $\Delta_{x}=\omega_{x}-\omega_{p}$ and
$\Delta_{K}=\omega_{K}-\omega_{p}$. To see the effect of the MPD clearly, it
is convenient to transform the Hamiltonian (15) to the squeezed frame by
performing a unitary squeezing transformation,
$\hat{U}_{s}=\exp[r_{p}(\hat{b}^{2}-\hat{b}^{\dagger 2})/2]$, where the
squeezing parameter $r_{p}$ is defined via
$r_{p}=(1/2)\operatorname{arctanh}\left(\Omega_{p}/\Delta_{x}\right)$,
yielding

$\begin{split}\frac{\hat{H}_{\mathrm{mm}}^{\mathrm{S}}}{\hbar}=&\Delta_{s}\hat{b}^{\dagger}_{s}\hat{b}_{s}+\Delta_{K}\hat{s}^{\dagger}\hat{s}+\big{[}g_{r}(\hat{b}_{s}\hat{s}^{\dagger}+\hat{b}_{s}^{\dagger}\hat{s}^{\dagger}e^{\mathrm{i}2\omega_{p}t})\\\
&-g_{c}(\hat{b}_{s}^{\dagger}\hat{s}^{\dagger}+\hat{b}_{s}\hat{s}^{\dagger}e^{\mathrm{i}2\omega_{p}t})+\mathrm{H.c.}\big{]},\end{split}$
(16)

where $\Delta_{s}=\Delta_{x}/\cosh(2r_{p})$ is the squeezed-phonon frequency,
$g_{r}=g\cosh(r_{p})$ and $g_{c}=g\sinh(r_{p})$ are the enhanced magnon-
mechanical coupling strengths. Such an enhancement is a consequence of the
MPD, which modifies the eigenstates of the mechanical Hamiltonian and
generates the squeezed phonons with the amplified fluctuations responsible for
the larger interaction with the magnon. In addition, Eq. (16) implies that
different quantum dynamical processes can be achieved under different
parameter regimes. As an example, we discuss the magnon-mechanical interaction
in the regime of $\Delta_{s}=\Delta_{K}$, where the magnon is in resonance
with the mechanical phonon (see Appendix A for more detailed discussions). The
unitary dynamics of such a system is governed by a Jaynes-Cummings-type
Hamiltonian

$\begin{split}\frac{\hat{H}_{\mathrm{mm,r}}^{\mathrm{S}}}{\hbar}\simeq&\Delta_{s}\hat{b}^{\dagger}_{s}\hat{b}_{s}+\Delta_{K}\hat{s}^{\dagger}\hat{s}+g_{r}(\hat{b}_{s}\hat{s}^{\dagger}+\mathrm{H.c.}),\end{split}$
(17)

which suggests a coherent exchange of the phonon excitation and the magnon,
with the coupling being exponentially enhanced. The direct realization of such
a coherent coupling between the magnon and the mechanical phonons could be
versatile for various applications, e.g., magnon-assisted ground-state cooling
of the mechanical motion Gonzalez-Ballestero  _et al._ (2020a), and quantum
control and detection of the magnon excitations by measuring the mechanical
motion Li  _et al._ (2021).

In addition to the coherent dynamics described by Eq. (17), we take into
account the couplings of the mechanical oscillator and the nanomagnet to the
Markovian baths, which cause the mechanics to be damped at a rate $\gamma_{x}$
and the magnon at a rate $\gamma_{s}$. In the presence of the MPD, the noise
coming from the mechanical bath is also amplified. In the weak mechanical
dissipation limit $\gamma_{x}\ll\Delta_{s}$, the mechanical bath with a
thermal occupation $\bar{n}_{x}$ corresponds to a bath for the $\hat{b}_{s}$
mode with a thermal occupation
$\bar{n}_{x}^{s}=\bar{n}_{x}\cosh(2r_{p})+\sinh^{2}(r_{p})$ (see Appendix B
for details). That is, the $\hat{b}_{s}$ mode can be considered as being
coupled, with an unchanged vacuum damping rate $\gamma_{x}$, to a thermal bath
but with thermal occupation $\bar{n}_{x}^{s}$. The system dynamics under
dissipation can then be described by the Lindblad master equation

$\dot{\hat{\rho}}=-\frac{\mathrm{i}}{\hbar}[\hat{H}_{\mathrm{mm}}^{\mathrm{S}},\hat{\rho}]+\mathcal{L}_{x}[\hat{\rho}]+\mathcal{L}_{s}[\hat{\rho}].$
(18)

Here, $\hat{H}_{\mathrm{mm}}^{\mathrm{S}}$ is given by Eq. (16), $\hat{\rho}$
is the density operator,
$\mathcal{L}_{x}[\hat{\rho}]=\gamma_{x}[\left(\bar{n}_{x}^{s}+1\right)L_{\hat{b}_{s}}\hat{\rho}+\bar{n}_{x}^{s}L_{\hat{b}_{s}^{\dagger}}\hat{\rho}]$
and
$\mathcal{L}_{s}[\hat{\rho}]=\gamma_{s}[\left(\bar{n}_{s}+1\right)L_{\hat{s}}\hat{\rho}+\bar{n}_{s}L_{\hat{s}^{\dagger}}\hat{\rho}]$,
where
$L_{\hat{o}}\hat{\rho}\equiv\hat{o}\hat{\rho}\hat{o}^{\dagger}-[\hat{o}^{\dagger}\hat{o},\hat{\rho}]/2$.
Moreover,
$\bar{n}_{j}=\left[\exp\left(\hbar\omega_{j}/k_{B}T_{j}\right)-1\right]^{-1}$
($j=x,s$), with $k_{B}$ the Boltzmann constant and $T_{j}$ the temperature of
the thermal bath of each degree of freedom. We assume that all modes couple
with the thermal reservoirs at a common temperature $T$. It is also convenient
to define the mechanical quality ($Q$) factor as
$Q_{x}=\omega_{x}/\gamma_{x}$. The experimental values of $Q_{x}$ exceeding
$10^{8}$ have been demonstrated recently in nanomechanical resonators Norte
_et al._ (2016); Tsaturyan  _et al._ (2017); Ghadimi  _et al._ (2018).
Regarding the magnon linewidth, we consider $\gamma_{s}=2\pi\times 1$ MHz for
the YIG material Zhang  _et al._ (2014); Tabuchi  _et al._ (2015); Shen  _et
al._ (2022).

Next, we explore the physical regimes where our proposed magnon-mechanical
system can be achieved using realistic parameters. We assume that the
dimensions of the micromechanical cantilever are $(\ell,w,t)=(4,0.1,0.02)$
$\mu$m. Then, the frequency and the effective mass of this mechanical
oscillator are estimated, respectively, as $\omega_{x}\approx
3.516\times\left(t/\ell^{2}\right)\sqrt{E/12\rho}\approx 2\pi\times 3.8$ MHz
and $M=\rho\ell wt/4\approx 7\times 10^{-18}$ kg, with the Young’s modulus
$E\approx 1.22\times 10^{12}\mathrm{~{}Pa}$ and the mass density $\rho\approx
3.52\times 10^{3}\mathrm{~{}kg/m^{3}}$ for diamond. For the YIG nanomagnet,
the sizes in the range of 10 $\leq R\leq$ 100 nm are considered hereafter, in
accordance with the approximations for deriving magnon eigenmodes Gonzalez-
Ballestero  _et al._ (2020b). The magnon frequency $\omega_{K}=|\gamma|B_{z}$
can be tuned by the external magnetic field $B_{z}$, and we apply a field
$B_{z}\approx 0.084$ T such that the spin Hamiltonian (3) is valid. This leads
to $\omega_{K}\approx 2\pi\times 2.35$ GHz. In addition, we consider a large
field gradient $b_{0}\approx 10^{7}$ T/m for the nanomagnet Mamin  _et al._
(2007, 2012); Poggio and Degen (2010). Once the parameters $b_{0}$, $R$ and
$x_{\mathrm{zpf}}$ are given, the magnon-mechanical coupling can be determined
via $g=b_{0}x_{\mathrm{zpf}}\mathcal{M}_{K}V/2\hbar$. We list the main system
parameters used for our numerical calculations in Table 1 (see Appendix C).

Figure 2: Relevant parameters and quantum dynamics of the magnon-mechanical
hybrid system. (a) Coupling-dissipation ratio $g_{r}/\gamma_{s}$ versus the
squeezing parameter $r_{p}$ for the YIG spheres with radii $R=10$ and 100 nm.
The dashed and dotted horizontal lines correspond to the bare coupling rates
$g$ for $R=10$ and 100 nm, respectively. (b) Cooperativity $C_{r}$ versus the
mechanical $Q$ factor $Q_{x}$ for $r_{p}=0$ (without the MPD) and 2.5. (c)
Enhancement of the coupling, $g_{r}/g$, and the cooperativity,
$C_{r}^{\prime}/C$, versus $r_{p}$. The shaded areas in (a) and (d) indicate
the SCR with $g_{r}/\gamma_{s}>1$. (d) Time evolution of the occupation
probabilities of the mean phonon number
$\langle\hat{b}_{s}^{\dagger}\hat{b}_{s}\rangle$ and magnon number
$\langle\hat{s}^{\dagger}\hat{s}\rangle$. The solid curves and hollow marks
are obtained by numerically solving the master equation (18) with the total
Hamiltonian (16) and the effective Hamiltonian (17), respectively. (e) Time
evolution of the occupation probabilities of
$\langle\hat{b}_{s}^{\dagger}\hat{b}_{s}\rangle$ (solid and dashed curves) and
$\langle\hat{s}^{\dagger}\hat{s}\rangle$ (dotted and dashed-dotted curves)
when considering a low mechanical $Q$ factor of $Q_{x}=10^{3}$. The solid and
dotted curves are obtained without using an auxiliary squeezed-vacuum
reservoir (SVR), while the dashed and dashed-dotted curves are obtained after
introducing an auxiliary SVR with matching parameters $r_{e}=r_{p}$ and
$\theta_{e}=\pi$. The coherent state transfer, which is corrupted by the
amplified mechanical noise, can be recovered by introducing the auxiliary SVR
that eliminates completely the amplified noise. Other parameters used are
given in the main text.

In Fig. 2(a), we plot the ratio $g_{r}/\gamma_{s}$ versus $r_{p}$ for two
values of the nanomagnet radius $R$. Because we always have
$\gamma_{x}\ll\gamma_{s}$ for typical values of $Q_{x}$, the ratio
$g_{r}/\gamma_{s}>1$ can thus be seen as an indication of the system entering
the SCR, whereas $g_{r}/\gamma_{s}\leq 1$ corresponds to the weak-coupling
regime (WCR). One can see from Fig. 2(a) that $g_{r}/\gamma_{s}$ increases
gradually as a function of $r_{p}$ due to the effect of the MPD. For a
relatively small $R$ of 10 nm, the current system cannot reach the SCR even
for $r_{p}=2.5$, whereas for a larger $R$ of 100 nm, the system can be
enhanced into the SCR as long as $r_{p}>0.93$. Remarkably, one can obtain
$g_{r}/|\Delta_{s}|>0.1$ for $r_{p}\geq 2.42$, indicating that the system
enters the UCR. Figure 2(b) shows the cooperativity
$C_{r}=4g_{r}^{2}/[\gamma_{x}\gamma_{s}(1+\bar{n}_{x}^{s})]$ versus the
mechanical $Q$ factor $Q_{x}$ ranging from $10^{5}$ to $10^{8}$ for different
$r_{p}$, considering $R=100$ nm and a cryogenic temperature $T=10$ mK
Tsaturyan  _et al._ (2017). The system resides in the high-cooperativity
($C>1$) regime even at a moderate $Q_{x}\sim 10^{5}$, and $C_{r}$ increases
significantly with increasing $Q_{x}$. In contrast, increasing $r_{p}$ leads
to the decrease in $C_{r}$, due to the amplified mechanical noise responsible
for the rapidly accumulated thermal phonons $\bar{n}_{x}^{s}$ with increasing
the parametric drive. Note that this is generally a prominent problem for
realizing coherent energy transfer in previous schemes, however it can be
circumvented in our proposal as verified in the following.

By numerically solving the master equation (18), the time evolution of the
occupation probabilities of the mean phonon and magnon numbers can be
evaluated. Figure 2(d) shows $\langle\hat{b}_{s}^{\dagger}\hat{b}_{s}\rangle$
(solid curve) and $\langle\hat{s}^{\dagger}\hat{s}\rangle$ (dashed curve)
versus the evolution time for $r_{p}=2.5$ and $Q_{x}=10^{8}$. The hollow marks
are obtained by solving the master equation (18) with the effective
Hamiltonian (17), and are in good agreement with the exact numerical results
(solid and dashed curves). The typical vacuum Rabi oscillations occur for
multiple periods, revealing the SCR that the system resides. Notably, we also
find that the coherent dynamics can hardly be interrupted in the presence of
the amplified mechanical noise, and this is shown to be valid at least for
$Q_{x}\geq 10^{5}$. For larger mechanical damping, e.g., $Q_{x}=10^{3}$, the
coherent dynamics of the system are corrupted drastically by the amplified
thermal noise, with the state transfer between the mechanical phonon and the
magnon being completely decoupled, as depicted in Fig. 2(e) (see the top two
curves). To be more specific, the enhancement of the mechanical noise induced
by the phonon squeezing can be effectively evaluated via
$\gamma_{x}^{\prime}/\gamma_{x}=\cosh(2r_{p})$, where $\gamma_{x}^{\prime}$ is
the effective damping rate of the $\hat{b}_{s}$ mode. The enhancement of the
magnon-mechanical coupling is $g_{r}/g=\cosh(r_{p})$. For $Q_{x}\geq 10^{5}$
or $\gamma_{x}\leq 2\pi\times 38$ Hz, the enhanced damping rate
$\gamma_{x}^{\prime}\leq 2\pi\times 2.8$ kHz, which is at least three orders
of magnitude smaller than the parametrically-enhanced coupling $g_{r}\approx
2\pi\times 4.23$ MHz. This allows the observation of the coherent state
transfer in the system. In contrast, for relatively low $Q_{x}=10^{3}$ or
$\gamma_{x}=2\pi\times 3.8$ kHz, the damping rate is effectively enhanced by
the phonon squeezing to $\gamma_{x}^{\prime}=2\pi\times 0.28$ MHz. In this
case, $g_{r}$ is just about 15 times larger than $\gamma_{x}^{\prime}$, and
therefore, the amplified noise corrupts the coherent state transfer in the
system. This may lead to a certain limitation of the proposed scheme in case
of the mechanical resonator possessing a extremely-low $Q$. To eliminate
completely the amplified noise from the mechanical bath, a possible strategy
is to inject an orthogonally squeezed vacuum, such that the bath of the
$\hat{b}_{s}$ mode remains in vacuum (see Appendix B for more details). The
coherent dynamics of the system can then be recovered, as verified in Fig.
2(e) (see the bottom two curves). Remarkably, the cooperativity $C_{r}$ after
introducing the auxiliary reservoir is renewed as $C_{r}^{\prime}\equiv
4g_{r}^{2}/[\gamma_{x}\gamma_{s}(1+\bar{n}_{x})]$, which can be
parametrically-enhanced by increasing $r_{p}$, as shown Fig. 2(c).

3.2 Quantum dynamics with enhanced magnon-spin coupling

Now let us integrate the NV spin into the abovementioned magnon-mechanical
system. The Hamiltonian describing the hybrid tripartite system, in the
rotating frame with $\omega_{p}$, is given by

$\begin{split}\frac{\hat{H}_{\mathrm{Hybrid}}}{\hbar}=&\Delta_{x}\hat{b}^{\dagger}\hat{b}+\Delta_{K}\hat{s}^{\dagger}\hat{s}+\frac{1}{2}\Delta_{\mathrm{NV}}\hat{\sigma}_{z}-\frac{\Omega_{p}}{2}(\hat{b}^{2}+\hat{b}^{\dagger
2})\\\
&+\big{[}g(\hat{b}\hat{s}^{\dagger}+\hat{b}^{\dagger}\hat{s}^{\dagger}e^{\mathrm{i}2\omega_{p}t})+\lambda(\hat{b}\hat{\sigma}_{+}+\hat{b}^{\dagger}\hat{\sigma}_{+}e^{\mathrm{i}2\omega_{p}t})\\\
&+\mathrm{H.c.}\big{]},\end{split}$ (19)

with $\Delta_{\mathrm{NV}}=\omega_{\mathrm{NV}}-\omega_{p}$. We use the
squeezing transformation $\hat{U}_{s}$ again to diagonalize the mechanics-only
parts in $\hat{H}_{\mathrm{Hybrid}}$, and the resulting Hamiltonian reads

$\begin{split}\frac{\hat{H}_{\mathrm{Hybrid}}^{\mathrm{S}}}{\hbar}=&\Delta_{s}\hat{b}^{\dagger}_{s}\hat{b}_{s}+\Delta_{K}\hat{s}^{\dagger}\hat{s}+\frac{1}{2}\Delta_{\mathrm{NV}}\hat{\sigma}_{z}\\\
&+\big{[}g_{r}(\hat{b}_{s}\hat{s}^{\dagger}+\hat{b}_{s}^{\dagger}\hat{s}^{\dagger}e^{\mathrm{i}2\omega_{p}t})\\\
&-g_{c}(\hat{b}_{s}^{\dagger}\hat{s}^{\dagger}+\hat{b}_{s}\hat{s}^{\dagger}e^{\mathrm{i}2\omega_{p}t})\\\
&+\lambda_{r}(\hat{b}_{s}\hat{\sigma}_{+}+\hat{b}_{s}^{\dagger}\hat{\sigma}_{+}e^{\mathrm{i}2\omega_{p}t})\\\
&-\lambda_{c}(\hat{b}_{s}^{\dagger}\hat{\sigma}_{+}+\hat{b}_{s}\hat{\sigma}_{+}e^{\mathrm{i}2\omega_{p}t})+\mathrm{H.c.}\big{]},\end{split}$
(20)

where $\lambda_{r}=\lambda\cosh(r_{p})$ and $\lambda_{c}=\lambda\sinh(r_{p})$
are the enhanced spin-mechanical couplings by the MPD. In the dispersive
regime with properly large detunings (see Appendix D), the squeezed phonon
mode $\hat{b}_{s}$ is only virtually excited, which allows us to adiabatically
eliminate the mechanical degree of freedom from the dynamics. We further
consider that the magnon frequency is very close to the NV spin frequency,
namely $\omega_{K}\approx\omega_{\mathrm{NV}}$, which can be achieved by
introducing an additional static magnetic field (with the opposite direction
to $\textbf{B}_{z}$) for the nanomagnet (see relevant realistic parameters
below). Furthermore, the magnetic gradient $b_{1}$ is adjusted such that the
bare spin-mechanical coupling $\lambda$ is nearly equal to the bare magnon-
mechanical coupling $g$. Given these settings, the effective Hamiltonian
describing the phonon-mediated magnon-spin interactions can then be written as
(see Appendix D)

$\frac{\hat{H}_{\mathrm{ms}}^{\mathrm{S}}}{\hbar}=\Delta_{k}\hat{s}^{\dagger}\hat{s}+\Delta_{n}\hat{\sigma}_{+}\hat{\sigma}_{-}+g_{ms}(\hat{s}^{\dagger}\hat{\sigma}_{-}+\hat{s}\hat{\sigma}_{+}),$
(21)

where $\Delta_{k}=g_{r}^{2}/(\Delta_{K}-\Delta_{s})$,
$\Delta_{n}=\lambda_{r}^{2}/(\Delta_{\mathrm{NV}}-\Delta_{s})$ are the
effective detunings, and
$g_{ms}=g_{r}\lambda_{r}/[\Delta_{K(\mathrm{NV})}-\Delta_{s}]$ is the
effective magnon-spin coupling rate. The Hamiltonian (21) indicates that we
have achieved an effective two-mode system where the magnon is resonantly
coupled to the NV spin with an enhanced coupling rate.

Under a realistic consideration, the dissipative dynamics of the tripartite
system is governed by the master equation

$\dot{\hat{\rho}}=-\frac{\mathrm{i}}{\hbar}[\hat{H}_{\mathrm{Hybrid}}^{\mathrm{S}},\hat{\rho}]+\mathcal{L}_{x}[\hat{\rho}]+\mathcal{L}_{K}[\hat{\rho}]+\mathcal{L}_{z}[\hat{\rho}],$
(22)

where $\hat{H}_{\mathrm{Hybrid}}^{\mathrm{S}}$ is given in Eq. (20), and
$\mathcal{L}_{z}[\hat{\rho}]=\gamma_{z}L_{\hat{\sigma}_{z}}\hat{\rho}$ with
$\gamma_{z}$ the single spin dephasing rate of the NV center. Experimentally,
the NV dephasing time $T_{2}=1/\gamma_{z}$ can readily reach several hundreds
of kilohertz at both room temperature and cryogenic temperature Ishikawa  _et
al._ (2012); Mamin  _et al._ (2013); Bar-Gill  _et al._ (2013);
Ovartchaiyapong  _et al._ (2014), and hereafter we consider $\gamma_{z}\approx
2\pi\times 1$ kHz. As for other experimental parameters, we recall the
scenario considered in Sec. 3.1, where relevant experimental parameters are
$\omega_{x}\approx 2\pi\times 3.8$ MHz, $Q_{x}=10^{8}$, and $g\approx
2\pi\times 0.69$ MHz for $R=100$ nm. For the NV spin, we apply a static
magnetic field $B_{z}\approx 0.068$ T, which gives
$\omega_{\mathrm{NV}}\approx 2\pi\times 0.96$ GHz. Furthermore, an additional
local field $B_{z}^{\prime}\approx 0.034$ T is introduced to adjust the
frequency of the nanomagnet, such that
$\omega_{K}=|\gamma|(B_{z}-B_{z}^{\prime})\approx\omega_{\mathrm{NV}}$ Zhang
_et al._ (2014). Regarding the bare coupling between the NV spin and the
mechanical oscillator, we consider $b_{1}\approx 4.56\times 10^{7}$ T/m,
resulting $\lambda\approx 2\pi\times 0.69$ MHz, namely $\lambda\approx g$.
Then, the system dynamics can be evaluated by exactly solving the master
equation (22).

Figure 3: Quantum dynamics of the hybrid tripartite system in the dispersive
regime. (a) Time evolution of the occupation probabilities of the mean phonon
number, the mean magnon numbers and the spin excited state. The solid curves
and hollow marks are obtained by numerically solving the master equation (22)
with the total Hamiltonian (20) and the effective Hamiltonian (21),
respectively. (b),(c) Time evolution of the occupation probability of the spin
excited state for different magnon dissipation rate $\gamma_{s}$ and
mechanical $Q$ factor $Q_{x}$. In (b) and (c), $Q_{x}$ and $\gamma_{s}$ are
fixed, respectively, to be $10^{8}$ and $2\pi\times 0.1$ MHz. Other specific
parameters used here are given in the main text.

The dynamics of the hybrid system is examined for a proper MPD and the
suitably large detunings, for example, $r_{p}=1.54$, $\Delta_{s}=-55g_{r}$ and
$\Delta_{K}=-45g_{r}$. The numerical results are shown in Fig. 3(a). Here, we
assume that the NV spin is initially in the excited state, while the
nanomagnet and the mechanical oscillator are in their vacuum states. The
evident vacuum Rabi oscillations between the nanomagnet and the NV spin with
excited-state population $\langle\hat{\sigma}_{+}\hat{\sigma}_{-}\rangle$ can
be observed clearly, whereas the mechanical oscillator is nearly unexcited
(namely, virtually excited) with
$\langle\hat{b}_{s}^{\dagger}\hat{b}_{s}\rangle\approx 0$ during the
evolution. This implies that the quantum state initially stored in the NV spin
can be transferred to the nanomagnet through the mechanical oscillator serving
as a quantum bus, and vice versa. In our simulations, the decay rate of the
magnon mode is set to be $\gamma_{s}=2\pi\times 0.1$ MHz. Experimentally, a
narrow magnon linewidth $<2\pi\times 0.6$ MHz has been demonstrated in a YIG
sphere Zhang  _et al._ (2016b), which could be more smaller when considering
an ultrapure YIG Rameshti  _et al._ (2022). For the relatively strong magnon
dissipation, the state transfer becomes more inefficient with increasing
$\gamma_{s}$, while it exhibits strong robustness against the mechanical
dissipation for $Q_{x}$ as low as $10^{4}$, as shown in Figs. 3(b) and 3(c).

4 Conclusion

In conclusion, we have presented an experimentally feasible method to realize
the strong magnon-mechanical and phonon-mediated magnon-spin couplings in a
hybrid tripartite system. We have shown that the magnon mode supported by the
nanomagnet can be coupled, in a strong and tunable way, to the phonon mode of
the micromechanical cantilever by introducing the parametric amplification of
the mechanical motion. With experimentally accessible parameters, this magnon-
mechanical coupling can readily reach the strong-coupling regime, thus
removing the obstacle to achieving the strong coupling between the two quantum
systems with a large energy difference. By further coupling an NV spin to the
mechanical motion and working in the dispersive regime, we have engineered an
efficient strategy for quantum state transfer between the nanomagnet and the
NV center. This provides a versatile platform for processing complicated
quantum information tasks where the NV spin and the magnon can both act as
collaborate quantum memories. Our proposal could also find other potential
applications such as quantum control and measurement of magnon excitations,
and the design of quantum transducers and sensors, etc.

###### Acknowledgements.

This work was supported by the National Natural Science Foundation of China
(Grant Nos. 12205256, 11935006, 11774086, 12247105, and 1217050862), the Henan
Provincial Science and Technology Research Project (Grant Nos. 232102221001,
232102210175), HNQSTIT project (Grant No. 2022112), and the Fundamental
Research Funds for the Central Universities (Grant No. 2023FRFK06012).

## References

  * Yuan  _et al._ (2022) H. Yuan, Y. Cao, A. Kamra, R. A. Duine, and P. Yan, Phys. Rep. 965, 1 (2022).
  * Zare Rameshti  _et al._ (2022) B. Zare Rameshti, S. Viola Kusminskiy, J. A. Haigh, K. Usami, D. Lachance-Quirion, Y. Nakamura, C.-M. Hu, H. X. Tang, G. E. Bauer, and Y. M. Blanter, Phys. Rep. 979, 1 (2022).
  * Barman  _et al._ (2021) A. Barman, G. Gubbiotti, S. Ladak, and et al., J. Phys.: Condens. Matter 33, 413001 (2021).
  * Li  _et al._ (2020a) Y. Li, W. Zhang, V. Tyberkevych, W.-K. Kwok, A. Hoffmann, and V. Novosad, J. Appl. Phys. 128, 130902 (2020a).
  * Kamra  _et al._ (2020) A. Kamra, W. Belzig, and A. Brataas, Appl. Phys. Lett. 117, 090501 (2020).
  * Harder  _et al._ (2021) M. Harder, B. M. Yao, Y. S. Gui, and C.-M. Hu, J. Appl. Phys. 129, 201101 (2021).
  * Lachance-Quirion  _et al._ (2019) D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami, and Y. Nakamura, Appl. Phys. Express 12, 070101 (2019).
  * Huebl  _et al._ (2013) H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein, A. Marx, R. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett. 111, 127003 (2013).
  * Tabuchi  _et al._ (2014) Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Phys. Rev. Lett. 113, 083603 (2014).
  * Zhang  _et al._ (2014) X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Phys. Rev. Lett. 113, 156401 (2014).
  * Osada  _et al._ (2016) A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, Phys. Rev. Lett. 116, 223601 (2016).
  * Zhang  _et al._ (2016a) X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, Phys. Rev. Lett. 117, 123605 (2016a).
  * Li  _et al._ (2019) Y. Li, T. Polakovic, Y.-L. Wang, J. Xu, S. Lendinez, Z. Zhang, J. Ding, T. Khaire, H. Saglam, R. Divan, J. Pearson, W.-K. Kwok, Z. Xiao, V. Novosad, A. Hoffmann, and W. Zhang, Phys. Rev. Lett. 123, 107701 (2019).
  * Hou and Liu (2019) J. T. Hou and L. Liu, Phys. Rev. Lett. 123, 107702 (2019).
  * Xu  _et al._ (2020) J. Xu, C. Zhong, X. Han, D. Jin, L. Jiang, and X. Zhang, Phys. Rev. Lett. 125, 237201 (2020).
  * Yuan  _et al._ (2020) H. Y. Yuan, P. Yan, S. Zheng, Q. Y. He, K. Xia, and M.-H. Yung, Phys. Rev. Lett. 124, 053602 (2020).
  * Tabuchi  _et al._ (2015) Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, and Y. Nakamura, Science 349, 405 (2015).
  * Lachance-Quirion  _et al._ (2017) D. Lachance-Quirion, Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, and Y. Nakamura, Sci. Adv. 3, e1603150 (2017).
  * Lachance-Quirion  _et al._ (2020) D. Lachance-Quirion, S. P. Wolski, Y. Tabuchi, S. Kono, K. Usami, and Y. Nakamura, Science 367, 425 (2020).
  * Zhang  _et al._ (2016b) X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Sci. Adv. 2, e1501286 (2016b).
  * Li  _et al._ (2018) J. Li, S.-Y. Zhu, and G. S. Agarwal, Phys. Rev. Lett. 121, 203601 (2018).
  * Yu  _et al._ (2020) M. Yu, H. Shen, and J. Li, Phys. Rev. Lett. 124, 213604 (2020).
  * Shen  _et al._ (2022) Z. Shen, G.-T. Xu, M. Zhang, Y.-L. Zhang, Y. Wang, C.-Z. Chai, C.-L. Zou, G.-C. Guo, and C.-H. Dong, Phys. Rev. Lett. 129, 243601 (2022).
  * Kurizki  _et al._ (2015) G. Kurizki, P. Bertet, Y. Kubo, K. Mølmer, D. Petrosyan, P. Rabl, and J. Schmiedmayer, Proc. Natl Acad. Sci. U.S.A. 112, 3866 (2015).
  * Hisatomi  _et al._ (2016) R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa, A. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura, Phys. Rev. B 93, 174427 (2016).
  * Zhu  _et al._ (2020a) N. Zhu, X. Zhang, X. Han, C.-L. Zou, C. Zhong, C.-H. Wang, L. Jiang, and H. X. Tang, Optica 7, 1291 (2020a).
  * Chai  _et al._ (2022) C.-Z. Chai, Z. Shen, Y.-L. Zhang, H.-Q. Zhao, G.-C. Guo, C.-L. Zou, and C.-H. Dong, Photon. Res. 10, 820 (2022).
  * Cao and Yan (2019) Y. Cao and P. Yan, Phys. Rev. B 99, 214415 (2019).
  * Colombano  _et al._ (2020) M. F. Colombano, G. Arregui, F. Bonell, N. E. Capuj, E. Chavez-Angel, A. Pitanti, S. O. Valenzuela, C. M. Sotomayor-Torres, D. Navarro-Urrios, and M. V. Costache, Phys. Rev. Lett. 125, 147201 (2020).
  * Wolski  _et al._ (2020) S. P. Wolski, D. Lachance-Quirion, Y. Tabuchi, S. Kono, A. Noguchi, K. Usami, and Y. Nakamura, Phys. Rev. Lett. 125, 117701 (2020).
  * Crescini  _et al._ (2020) N. Crescini, C. Braggio, G. Carugno, A. Ortolan, and G. Ruoso, Appl. Phys. Lett. 117, 144001 (2020).
  * Lu  _et al._ (2021) T.-X. Lu, H. Zhang, Q. Zhang, and H. Jing, Phys. Rev. A 103, 063708 (2021).
  * Gonzalez-Ballestero  _et al._ (2020a) C. Gonzalez-Ballestero, J. Gieseler, and O. Romero-Isart, Phys. Rev. Lett. 124, 093602 (2020a).
  * Gonzalez-Ballestero  _et al._ (2020b) C. Gonzalez-Ballestero, D. Hümmer, J. Gieseler, and O. Romero-Isart, Phys. Rev. B 101, 125404 (2020b).
  * Holanda  _et al._ (2018) J. Holanda, D. Maior, A. Azevedo, and S. Rezende, Nature Phys. 14, 500 (2018).
  * Bozhko  _et al._ (2020) D. A. Bozhko, V. I. Vasyuchka, A. V. Chumak, and A. A. Serga, Low Temp. Phys. 46, 383 (2020).
  * Neuman  _et al._ (2020) T. c. v. Neuman, D. S. Wang, and P. Narang, Phys. Rev. Lett. 125, 247702 (2020).
  * Wang  _et al._ (2021) D. S. Wang, T. Neuman, and P. Narang, J. Phys. Chem. C 125, 6222 (2021).
  * Skogvoll  _et al._ (2021) I. C. Skogvoll, J. Lidal, J. Danon, and A. Kamra, Phys. Rev. Appl. 16, 064008 (2021).
  * Fukami  _et al._ (2021) M. Fukami, D. R. Candido, D. D. Awschalom, and M. E. Flatté, PRX Quantum 2, 040314 (2021).
  * Xiong  _et al._ (2022) W. Xiong, M. Tian, G.-Q. Zhang, and J. Q. You, Phys. Rev. B 105, 245310 (2022).
  * Lü  _et al._ (2015) X.-Y. Lü, Y. Wu, J. R. Johansson, H. Jing, J. Zhang, and F. Nori, Phys. Rev. Lett. 114, 093602 (2015).
  * Qin  _et al._ (2019) W. Qin, V. Macrì, A. Miranowicz, S. Savasta, and F. Nori, Phys. Rev. A 100, 062501 (2019).
  * Qin  _et al._ (2018) W. Qin, A. Miranowicz, P.-B. Li, X.-Y. Lü, J. Q. You, and F. Nori, Phys. Rev. Lett. 120, 093601 (2018).
  * Leroux  _et al._ (2018) C. Leroux, L. C. G. Govia, and A. A. Clerk, Phys. Rev. Lett. 120, 093602 (2018).
  * Chen  _et al._ (2021) Y.-H. Chen, W. Qin, X. Wang, A. Miranowicz, and F. Nori, Phys. Rev. Lett. 126, 023602 (2021).
  * Tang  _et al._ (2022) L. Tang, J. Tang, M. Chen, F. Nori, M. Xiao, and K. Xia, Phys. Rev. Lett. 128, 083604 (2022).
  * Li  _et al._ (2020b) P.-B. Li, Y. Zhou, W.-B. Gao, and F. Nori, Phys. Rev. Lett. 125, 153602 (2020b).
  * Wang  _et al._ (2022) Y. Wang, J.-L. Wu, J.-X. Han, Y. Xia, Y.-Y. Jiang, and J. Song, Phys. Rev. Applied 17, 024009 (2022).
  * Lemonde  _et al._ (2016) M.-A. Lemonde, N. Didier, and A. A. Clerk, Nat. Commun. 7, 1 (2016).
  * Zhao  _et al._ (2020) W. Zhao, S.-D. Zhang, A. Miranowicz, and H. Jing, Sci. China Phys. Mech. Astron. 63, 224211 (2020).
  * Liu  _et al._ (2023) J.-X. Liu, Y.-F. Jiao, Y. Li, X.-W. Xu, Q.-Y. He, and H. Jing, Sci. China Phys. Mech. Astron. 66, 230312 (2023).
  * Wang  _et al._ (2019) Y. Wang, C. Li, E. M. Sampuli, J. Song, Y. Jiang, and Y. Xia, Phys. Rev. A 99, 023833 (2019).
  * Zhu  _et al._ (2020b) C. J. Zhu, L. L. Ping, Y. P. Yang, and G. S. Agarwal, Phys. Rev. Lett. 124, 073602 (2020b).
  * Groszkowski  _et al._ (2020) P. Groszkowski, H.-K. Lau, C. Leroux, L. C. G. Govia, and A. A. Clerk, Phys. Rev. Lett. 125, 203601 (2020).
  * Wang  _et al._ (2020a) Y. Wang, J.-L. Wu, J. Song, Z.-J. Zhang, Y.-Y. Jiang, and Y. Xia, Phys. Rev. A 101, 053826 (2020a).
  * Wang  _et al._ (2020b) Y. Wang, J.-L. Wu, J.-X. Han, Y.-Y. Jiang, Y. Xia, and J. Song, Phys. Rev. A 102, 032601 (2020b).
  * Qin  _et al._ (2021) W. Qin, A. Miranowicz, H. Jing, and F. Nori, Phys. Rev. Lett. 127, 093602 (2021).
  * (59) M. Villiers, W. C. Smith, A. Petrescu, A. Borgognoni, M. Delbecq, A. Sarlette, M. Mirrahimi, P. Campagne-Ibarcq, T. Kontos, and Z. Leghtas, arXiv:2212.04991 .
  * Ge  _et al._ (2019) W. Ge, B. C. Sawyer, J. W. Britton, K. Jacobs, J. J. Bollinger, and M. Foss-Feig, Phys. Rev. Lett. 122, 030501 (2019).
  * Burd  _et al._ (2021) S. C. Burd, R. Srinivas, H. M. Knaack, W. Ge, A. C. Wilson, D. J. Wineland, D. Leibfried, J. J. Bollinger, D. Allcock, and D. Slichter, Nat. Phys. 17, 898 (2021).
  * Affolter  _et al._ (2023) M. Affolter, W. Ge, B. Bullock, S. C. Burd, K. A. Gilmore, J. F. Lilieholm, A. L. Carter, and J. J. Bollinger, Phys. Rev. A 107, 032425 (2023).
  * Pan  _et al._ (2023) X.-F. Pan, X.-L. Hei, X.-L. Dong, J.-Q. Chen, C.-P. Shen, H. Ali, and P.-B. Li, Phys. Rev. A 107, 023722 (2023).
  * Hei  _et al._ (2023) X.-L. Hei, P.-B. Li, X.-F. Pan, and F. Nori, Phys. Rev. Lett. 130, 073602 (2023).
  * Asjad  _et al._ (2023) M. Asjad, J. Li, S.-Y. Zhu, and J. You, Fundam. Res. 3, 3 (2023).
  * Sharma  _et al._ (2021) S. Sharma, V. A. S. V. Bittencourt, A. D. Karenowska, and S. V. Kusminskiy, Phys. Rev. B 103, L100403 (2021).
  * Sun  _et al._ (2021) F.-X. Sun, S.-S. Zheng, Y. Xiao, Q. Gong, Q. He, and K. Xia, Phys. Rev. Lett. 127, 087203 (2021).
  * Vinante  _et al._ (2011) A. Vinante, G. Wijts, O. Usenko, L. Schinkelshoek, and T. Oosterkamp, Nat. Commun. 2, 1 (2011).
  * Burgess  _et al._ (2013) J. Burgess, A. Fraser, F. F. Sani, D. Vick, B. Hauer, J. Davis, and M. Freeman, Science 339, 1051 (2013).
  * Vinante and Falferi (2013) A. Vinante and P. Falferi, Phys. Rev. Lett. 111, 207203 (2013).
  * den Haan  _et al._ (2015) A. M. J. den Haan, J. J. T. Wagenaar, J. M. de Voogd, G. Koning, and T. H. Oosterkamp, Phys. Rev. B 92, 235441 (2015).
  * Kolkowitz  _et al._ (2012) S. Kolkowitz, A. C. B. Jayich, Q. P. Unterreithmeier, S. D. Bennett, P. Rabl, J. Harris, and M. D. Lukin, Science 335, 1603 (2012).
  * Rugar and Grütter (1991) D. Rugar and P. Grütter, Phys. Rev. Lett. 67, 699 (1991).
  * Szorkovszky  _et al._ (2011) A. Szorkovszky, A. C. Doherty, G. I. Harris, and W. P. Bowen, Phys. Rev. Lett. 107, 213603 (2011).
  * Rabl  _et al._ (2009) P. Rabl, P. Cappellaro, M. V. G. Dutt, L. Jiang, J. R. Maze, and M. D. Lukin, Phys. Rev. B 79, 041302 (2009).
  * Rabl  _et al._ (2010) P. Rabl, S. J. Kolkowitz, F. Koppens, J. Harris, P. Zoller, and M. D. Lukin, Nat. Phys. 6, 602 (2010).
  * Childress  _et al._ (2006) L. Childress, M. Gurudev Dutt, J. Taylor, A. Zibrov, F. Jelezko, J. Wrachtrup, P. Hemmer, and M. Lukin, Science 314, 281 (2006).
  * Taylor  _et al._ (2008) J. M. Taylor, P. Cappellaro, L. Childress, L. Jiang, D. Budker, P. Hemmer, A. Yacoby, R. Walsworth, and M. Lukin, Nature Phys. 4, 810 (2008).
  * Walker (1957) L. R. Walker, Phys. Rev. 105, 390 (1957).
  * Walker (1958) L. R. Walker, J. Appl. Phys. 29, 318 (1958).
  * Röschmann and Dötsch (1977) P. Röschmann and H. Dötsch, Phys. Status Solidi B 82, 11 (1977).
  * Mills (2006) D. Mills, J. Magn. Magn. Mater. 306, 16 (2006).
  * Kittel (1948) C. Kittel, Phys. Rev. 73, 155 (1948).
  * Mamin  _et al._ (2012) H. J. Mamin, C. T. Rettner, M. H. Sherwood, L. Gao, and D. Rugar, Appl. Phys. Lett. 100, 013102 (2012).
  * Li  _et al._ (2016) P.-B. Li, Z.-L. Xiang, P. Rabl, and F. Nori, Phys. Rev. Lett. 117, 015502 (2016).
  * Sánchez Muñoz  _et al._ (2018) C. Sánchez Muñoz, A. Lara, J. Puebla, and F. Nori, Phys. Rev. Lett. 121, 123604 (2018).
  * Zhou  _et al._ (2022) Y. Zhou, C.-S. Hu, D.-Y. Lü, X.-K. Li, H.-M. Huang, Y.-C. Xiong, and X.-Y. Lü, Photon. Res. 10, 1640 (2022).
  * Li  _et al._ (2021) J. Li, Y.-P. Wang, W.-J. Wu, S.-Y. Zhu, and J. You, PRX Quantum 2, 040344 (2021).
  * Norte  _et al._ (2016) R. A. Norte, J. P. Moura, and S. Gröblacher, Phys. Rev. Lett. 116, 147202 (2016).
  * Tsaturyan  _et al._ (2017) Y. Tsaturyan, A. Barg, E. S. Polzik, and A. Schliesser, Nat. Nanotechnol. 12, 776 (2017).
  * Ghadimi  _et al._ (2018) A. H. Ghadimi, S. A. Fedorov, N. J. Engelsen, M. J. Bereyhi, R. Schilling, D. J. Wilson, and T. J. Kippenberg, Science 360, 764 (2018).
  * Mamin  _et al._ (2007) H. Mamin, M. Poggio, C. Degen, and D. Rugar, Nat. Nanotechnol. 2, 301 (2007).
  * Poggio and Degen (2010) M. Poggio and C. L. Degen, Nanotechnology 21, 342001 (2010).
  * Ishikawa  _et al._ (2012) T. Ishikawa, K.-M. C. Fu, C. Santori, V. M. Acosta, R. G. Beausoleil, H. Watanabe, S. Shikata, and K. M. Itoh, Nano Lett. 12, 2083 (2012).
  * Mamin  _et al._ (2013) H. Mamin, M. Kim, M. Sherwood, C. Rettner, K. Ohno, D. Awschalom, and D. Rugar, Science 339, 557 (2013).
  * Bar-Gill  _et al._ (2013) N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, and R. L. Walsworth, Nat. Commun. 4, 1 (2013).
  * Ovartchaiyapong  _et al._ (2014) P. Ovartchaiyapong, K. W. Lee, B. A. Myers, and A. C. B. Jayich, Nat. Commun. 5, 1 (2014).
  * Rameshti  _et al._ (2022) B. Z. Rameshti, S. V. Kusminskiy, J. A. Haigh, K. Usami, D. Lachance-Quirion, Y. Nakamura, C.-M. Hu, H. X. Tang, G. E. Bauer, and Y. M. Blanter, Phys. Rep. 979, 1 (2022).
  * Planat  _et al._ (2020) L. Planat, A. Ranadive, R. Dassonneville, J. Puertas Martínez, S. Léger, C. Naud, O. Buisson, W. Hasch-Guichard, D. M. Basko, and N. Roch, Phys. Rev. X 10, 021021 (2020).
  * Grebel  _et al._ (2021) J. Grebel, A. Bienfait, E. Dumur, H.-S. Chang, M.-H. Chou, C. R. Conner, G. A. Peairs, R. G. Povey, Y. P. Zhong, and A. N. Cleland, Appl. Phys. Lett. 118, 142601 (2021).

Appendix A Derivation of the coherent coupling between the magnon and the
mechanical phonon

Generally, the direct realization of the coherent coupling between the magnon
and the mechanical phonon is difficult due to their large difference in the
energy scales. We overcome this problem by choosing a proper parameter regime,
i.e., $\Delta_{s}=\Delta_{K}$. To clarify this, we first give the system
Hamiltonian (16) in the interaction picture after setting
$\Delta_{s}=\Delta_{K}$,

$\begin{split}\frac{\hat{H}_{\mathrm{mm}}^{\mathrm{S,I}}}{\hbar}=&g_{r}\hat{b}_{s}\hat{s}^{\dagger}-g_{c}\hat{b}_{s}^{\dagger}\hat{s}^{\dagger}e^{\mathrm{i}2\Delta_{s}t}+g_{r}\hat{b}_{s}^{\dagger}\hat{s}^{\dagger}e^{\mathrm{i}2(\omega_{p}+\Delta_{s})t}\\\
&-g_{c}\hat{b}_{s}\hat{s}^{\dagger}e^{\mathrm{i}2\omega_{p}t}+\mathrm{H.c.}.\end{split}$
(23)

Then, the contributions of each terms in the above Hamiltonian to the system
dynamics can be easily identified by comparing the corresponding detunings. In
the regime of $\Delta_{s}=\Delta_{K}$, the system parameters satisfy
$2\sinh^{2}(r_{p})\omega_{p}=\cosh(2r_{p})\omega_{K}-\omega_{x}$, which is the
exact condition for resonance. Based on this, we plot in Fig. 4 the ratios of
$2|\Delta_{s}|$, $2(\omega_{p}+\Delta_{s})$ and $2\omega_{p}$ over
$g_{\mathrm{max}}\equiv\max\\{g_{r},g_{c}\\}$ versus $r_{p}\in[0,3]$. The
results show that we can safely neglect the contributions of the three terms
that oscillate rapidly by the rotating-wave approximation (RWA), and keep only
the slowly oscillating term
$\propto(\hat{b}_{s}\hat{s}^{\dagger}+\mathrm{H.c.})$, at least for $r_{p}\leq
2.5$. Furthermore, in the weak parametric drive region of relatively small
$r_{p}$, the resonance condition makes $\omega_{p}$ and the consequent
detunings extremely large, which may exceed the range of experimentally
feasible parameters. Nonetheless, this can be avoided when working with a
strong parametric drive that is considered in this work for a large coupling
enhancement. The system Hamiltonian (16) is thus reduced to

$\begin{split}\frac{\hat{H}_{\mathrm{mm,r}}^{\mathrm{S}}}{\hbar}\simeq&\Delta_{s}\hat{b}_{s}^{\dagger}\hat{b}_{s}+\Delta_{K}\hat{s}^{\dagger}\hat{s}+g_{r}(\hat{b}_{s}\hat{s}^{\dagger}+\mathrm{H.c.}),\end{split}$
(24)

A direct demonstration of the validity of Eq. (24) is given in Fig. 2(d),
where the effective Hamiltonian (24) reproduces exactly the system dynamics
governed by the total Hamiltonian (16) for $r_{p}=2.5$.

Figure 4: Frequency detunings over coupling versus $r_{p}$. The shaded area
shows the invalid region for the RWA with ratios less than 10. The parameters
are the same as in Fig. 2(a).

Appendix B Master equation and squeezing-induced noise

1 Derivation of the master equation

Considering the dissipation of the mechanical mode to a Markovian bath of
thermal occupation $\bar{n}_{x}$ and damping rate $\gamma_{x}$, as well as the
magnon mode to a bath of thermal occupation $\bar{n}_{s}$ and damping rate
$\gamma_{s}$, the dynamics of the hybrid magnon-mechanical system, in the
absence of the MPD, is described by the following master equation

$\begin{split}\dot{\hat{\rho}}=&-\frac{\mathrm{i}}{\hbar}[\hat{H}_{\mathrm{mm}}^{\prime},\hat{\rho}]+\gamma_{x}\left(\bar{n}_{x}+1\right)L_{\hat{b}}\hat{\rho}+\gamma_{x}\bar{n}_{x}L_{\hat{b}^{\dagger}}\hat{\rho}\\\
&+\gamma_{s}\left(\bar{n}_{s}+1\right)L_{\hat{s}}\hat{\rho}+\gamma_{s}\bar{n}_{s}L_{\hat{s}^{\dagger}}\hat{\rho},\end{split}$
(25)

where $\hat{H}_{\mathrm{mm}}^{\prime}$ is given by Eq. (15) with
$\Omega_{p}=\omega_{p}=0$, $\hat{\rho}$ is the density operator, and
$L_{\hat{o}}\hat{\rho}\equiv\hat{o}\hat{\rho}\hat{o}^{\dagger}-[\hat{o}^{\dagger}\hat{o},\hat{\rho}]/2$.

After introducing the MPD, the mechanical phonons can be squeezed, resulting
in a squeezed-phonon mode
$\hat{b}_{s}=\hat{b}\cosh(r_{p})+\hat{b}^{\dagger}\sinh(r_{p})$. In terms of
the mode $\hat{b}_{s}$, the master equation (25) can be rewritten as

$\begin{split}\dot{\hat{\rho}}=&-\frac{\mathrm{i}}{\hbar}[\hat{H}_{\mathrm{mm}}^{\mathrm{S}},\hat{\rho}]+\gamma_{x}\left(\bar{n}_{x}^{s}+1\right)L_{\hat{b}_{s}}\hat{\rho}+\gamma_{x}\bar{n}_{x}^{s}L_{\hat{b}_{s}^{\dagger}}\hat{\rho}\\\
&-\gamma_{x}M(2\bar{n}_{x}+1)\mathscr{L}_{\hat{b}_{s}}\hat{\rho}-\gamma_{x}M(2\bar{n}_{x}+1)\mathscr{L}_{\hat{b}_{s}^{\dagger}}\hat{\rho}\\\
&+\gamma_{s}\left(\bar{n}_{s}+1\right)L_{\hat{s}}\hat{\rho}+\gamma_{s}\bar{n}_{s}L_{\hat{s}^{\dagger}}\hat{\rho},\end{split}$
(26)

where $\hat{H}_{\mathrm{mm}}^{\mathrm{S}}$ is given by Eq. (16),
$\bar{n}_{x}^{s}=\bar{n}_{x}\cosh(2r_{p})+\sinh^{2}(r_{p})$,
$M=\cosh(r_{p})\sinh(r_{p})$, and
$\mathscr{L}_{\hat{o}}\hat{\rho}\equiv\hat{o}\hat{\rho}\hat{o}-[\hat{o}\hat{o},\hat{\rho}]/2$.
Equation (26) shows that when the mechanical mode $\hat{b}$ is coupled to a
Markovian bath of thermal occupancy $\bar{n}_{x}$, the $\hat{b}_{s}$ mode is
driven by a squeezed reservoir of thermal population $\bar{n}_{x}^{s}$.

In the limit of a weak mechanical dissipation, i.e.,
$\gamma_{x}\sinh(2r_{p})\ll\Delta_{s}$, the two terms (associated with
$\mathcal{L}_{\hat{b}_{s}}$ and $\mathcal{L}_{\hat{b}_{s}^{\dagger}}$)
describing the two-phonon correlation in Eq. (26) are high-frequency
oscillating terms and can thus be neglected. Therefore, the master equation
(26) is reduced to

$\begin{split}\dot{\hat{\rho}}=&-\frac{\mathrm{i}}{\hbar}[\hat{H}_{\mathrm{mm}}^{\mathrm{S}},\hat{\rho}]+\gamma_{x}\left(\bar{n}_{x}^{s}+1\right)L_{\hat{b}_{s}}\hat{\rho}+\gamma_{x}\bar{n}_{x}^{s}L_{\hat{b}_{s}^{\dagger}}\hat{\rho}\\\
&+\gamma_{s}\left(\bar{n}_{s}+1\right)L_{\hat{s}}\hat{\rho}+\gamma_{s}\bar{n}_{s}L_{\hat{s}^{\dagger}}\hat{\rho},\end{split}$
(27)

which is exactly the master equation (18) in the main text. From Eq. (27), we
find that the squeezed-phonon mode $\hat{b}_{s}$ can be considered as being
coupled, with an unchanged vacuum damping rate $\gamma_{x}$, to a thermal bath
but with an amplified thermal occupation $\bar{n}_{x}^{s}$. Apparently, the
thermal noise with thermal occupation $\bar{n}_{x}^{s}$ increases drastically
with increasing the MPD. In general, this amplified thermal noise could
corrupt the coherent dynamics of the system even with an exponentially
enhanced coupling, thus limiting the application of the system in the quantum
regime. However, it can hardly affect the coherent state transfer in our
system at least for $Q\geq 10^{5}$, as the amplified thermal noise remains far
less than the enhanced coupling strength even for a strong MPD (e.g.,
$r_{p}=2.5$). We note that although the intrinsic mechanical damping can be
extremely low in state-of-the-art experiments, e.g., $Q_{x}\geq 10^{8}$ in
nanomechanical resonators Norte  _et al._ (2016); Tsaturyan  _et al._ (2017);
Ghadimi  _et al._ (2018), our scheme presents a clear advantage in relaxing
the high-$Q$ requirement of the mechanical resonator in realistic experiments.

2 Elimination of the squeezing-induced noise

To eliminate the amplified noise from the mechanical bath, a possible strategy
is to use the squeezed-vacuum-reservoir technique, which has been widely used
in previous schemes Lü  _et al._ (2015); Qin  _et al._ (2019, 2018); Leroux
_et al._ (2018); Chen  _et al._ (2021); Tang  _et al._ (2022); Li  _et al._
(2020b); Wang  _et al._ (2022). The technique was first proposed in an
optomechanical system to eliminate the noise of the squeezed cavity mode Lü
_et al._ (2015). It employs an auxiliary, broadband squeezed-vacuum field to
drive the cavity, which is phase matched with the parametric drive that
squeezes the cavity mode. This ensures that the squeezed cavity mode is
equivalently coupled to a vacuum bath, thereby allowing one to describe the
system dynamics with a simplified master equation in the standard Lindblad
form.

To demonstrate more explicitly how to eliminate the amplified mechanical noise
in our scheme using the above auxiliary-reservoir technique, we now derive the
master equation for the hybrid magnon-mechanical system. We first assume that
the mechanical mode is coupled to a squeezed thermal reservoir with a
squeezing parameter $r_{e}$ and a reference phase $\theta_{e}$. The dynamics
of the system, in the absence of the MPD, is described by the master equation

$\begin{split}\dot{\hat{\rho}}=&-\frac{\mathrm{i}}{\hbar}[\hat{H}_{\mathrm{mm}}^{\prime},\hat{\rho}]+\gamma_{x}\left(\mathcal{N}+1\right)L_{\hat{b}}\hat{\rho}+\gamma_{x}\mathcal{N}L_{\hat{b}^{\dagger}}\hat{\rho}\\\
&-\gamma_{x}\mathcal{M}\mathscr{L}_{\hat{b}}\hat{\rho}-\gamma_{x}\mathcal{M}^{\ast}\mathscr{L}_{\hat{b}^{\dagger}}\hat{\rho}\\\
&+\gamma_{s}\left(\bar{n}_{s}+1\right)L_{\hat{s}}\hat{\rho}+\gamma_{s}\bar{n}_{s}L_{\hat{s}^{\dagger}}\hat{\rho},\end{split}$
(28)

where $\mathcal{N}$ and $\mathcal{M}$ describe the thermal noise and the two-
phonon correlations caused by the squeezed reservoir, which are given,
respectively, by

$\displaystyle\mathcal{N}$
$\displaystyle=\bar{n}_{x}\cosh\left(2r_{e}\right)+\sinh^{2}\left(r_{e}\right),$
(29) $\displaystyle\mathcal{M}$
$\displaystyle=\left(2\bar{n}_{x}+1\right)\cosh\left(r_{e}\right)\sinh\left(r_{e}\right)e^{\mathrm{i}\theta_{e}}.$

When squeezed, the master equation (28) can be rewritten, in terms of the mode
$\hat{b}_{s}$, as

$\begin{split}\dot{\hat{\rho}}=&-\frac{\mathrm{i}}{\hbar}[\hat{H}_{\mathrm{mm}}^{\mathrm{S}},\hat{\rho}]+\gamma_{x}\left(\mathcal{N}_{s}+1\right)L_{\hat{b}_{s}}\hat{\rho}+\gamma_{x}\mathcal{N}_{s}L_{\hat{b}_{s}^{\dagger}}\hat{\rho}\\\
&-\gamma_{x}\mathcal{M}_{s}\mathscr{L}_{\hat{b}_{s}}\hat{\rho}-\gamma_{x}\mathcal{M}_{s}^{\ast}\mathscr{L}_{\hat{b}^{\dagger}_{s}}\hat{\rho}\\\
&+\gamma_{s}\left(\bar{n}_{s}+1\right)L_{\hat{s}}\hat{\rho}+\gamma_{s}\bar{n}_{s}L_{\hat{s}^{\dagger}}\hat{\rho},\end{split}$
(30)

where $\mathcal{N}_{s}$ and $\mathcal{M}_{s}$ are given, respectively, by

$\displaystyle\mathcal{N}_{s}=$
$\displaystyle\left[\bar{n}_{x}\cosh\left(2r_{e}\right)\\!+\\!\sinh^{2}\left(r_{e}\right)\right]\cosh\left(2r_{p}\right)\\!+\\!\sinh^{2}\left(r_{p}\right)$
(31)
$\displaystyle+(\bar{n}_{x}\\!+\\!\frac{1}{2})\sinh\left(2r_{e}\right)\sinh\left(2r_{p}\right)\cos\left(\theta_{e}\right),$
$\displaystyle\mathcal{M}_{s}=$
$\displaystyle-\left(2\bar{n}_{x}\\!+\\!1\right)\big{\\{}\frac{1}{2}\sinh\left(2r_{p}\right)\cosh\left(2r_{e}\right)\\!+\\!\frac{1}{2}\sinh\left(2r_{e}\right)$
$\displaystyle\times\\!\left[\exp\left(\mathrm{i}\theta_{e}\right)\cosh^{2}\left(r_{p}\right)\\!+\\!\exp\left(-\mathrm{i}\theta_{e}\right)\sinh^{2}\left(r_{p}\right)\right]\big{\\}},$

corresponding to the effective thermal noise and two-phonon correlations of
the squeezed phonon mode. These two terms describe the undesired noise induced
by the phonon squeezing, which can be eliminated by setting proper parameters
of the squeezed reservoir. When choosing $r_{e}=r_{p}$ and $\theta_{e}=\pm
n\pi$ ($n=1,3,5,\dots$), $\mathcal{N}_{s}$ and $\mathcal{M}_{s}$ can be
reduced to $\bar{n}_{x}$ and 0, respectively. The master equation (30) is thus
simplified to

$\begin{split}\dot{\hat{\rho}}=&-\frac{\mathrm{i}}{\hbar}[\hat{H}_{\mathrm{mm}}^{\mathrm{S}},\hat{\rho}]+\gamma_{x}\left(\bar{n}_{x}+1\right)L_{\hat{b}_{s}}\hat{\rho}+\gamma_{x}\bar{n}_{x}L_{\hat{b}_{s}^{\dagger}}\hat{\rho}\\\
&+\gamma_{s}\left(\bar{n}_{s}+1\right)L_{\hat{s}}\hat{\rho}+\gamma_{s}\bar{n}_{s}L_{\hat{s}^{\dagger}}\hat{\rho}.\end{split}$
(32)

Apparently, the squeezing-induced noises are completely eliminated, and the
mode $\hat{b}_{s}$ is equivalently coupled to a thermal bath with the original
thermal occupation $\bar{n}_{x}$.

Table 1: System parameters. Unless otherwise specified, these parameters are applied to the numerical calculations in the main text. Symbol |  Name |  Value
---|---|---
$\omega_{x}$ |  Mechanical resonance frequency |  $2\pi\times 3.8$ MHz
$Q_{x}$ |  Mechanical quality factor |  $10^{8}$
$\gamma_{x}$ |  Mechanical linewidth |  $2\pi\times 0.038$ Hz
$\bar{n}_{x}$ |  Thermal phonon occupation |  55
$(\ell,w,t)$ |  Dimensions of micromechanical cantilever |  (4,0.1,0.02)$\mu$m
$M$ |  Effective mass of cantilever |  $7.0\times 10^{-18}$ kg
$x_{\mathrm{zpf}}$ |  Zero-point fluctuation amplitude |  $2.69\times 10^{-13}$ m
$\omega_{p}$ |  Mechanical parametric drive frequency |  $2\pi\times 2.382$ GHz
$\Omega_{p}$ |  Mechanical parametric drive amplitude |  $2\pi\times 2.378$ GHz
$\omega_{K}$ |  Kittel magnon frequency |  $2\pi\times 2.35$ GHz
$\gamma_{s}$ |  Magnon linewidth |  $2\pi\times 1.0$ MHz
$\bar{n}_{s}$ |  Thermal magnon occupation |  $1.3\times 10^{-5}$
$T$ |  Mechanical and magnon bath temperatures |  10 mK
$R$ |  Radius of YIG nanosphere |  100 nm
$V$ |  Volume of YIG nanosphere |  $4.2\times 10^{-21}$ m3
$M_{m}$ |  Effective mass of YIG nanosphere |  $2.17\times 10^{-18}$ kg
$M_{t}$ |  Effective mass of Dy tip |  $2.24\times 10^{-18}$ kg
$\omega_{\mathrm{NV}}$ |  NV spin frequency |  $2\pi\times 2.35$ GHz
$\gamma_{z}$ |  Single NV spin dephasing rate |  $2\pi\times 1.0$ kHz
$b_{0}$ |  Gradient of external magnetic field |  $10^{7}$ T/m
$b_{1}$ |  Magnetic field gradient produced by Dy tip |  $4.5\times 10^{7}$ T/m
$B_{z}$ |  Homogeneous component of external magnetic field |  0.084 T
$g(\lambda)$ |  Magnon(Spin)-mechanical coupling rate |  $2\pi\times 0.69$ MHz

Regarding the experimental implementation of the above approach, a possible
arrangement is to integrate a microwave optomechanical system into the present
hybrid setup, as discussed detailedly in Ref. Wang  _et al._ (2022). A
microwave resonator, which is electrically coupled to the mechanical
resonator, is terminated by a superconducting quantum interference device
(SQUID) that provides squeezed microwave inputs to the mechanical mode. The
microwave drive with a sufficiently large squeezing bandwidth (experimental
values of about gigahertz have been reported Planat  _et al._ (2020); Grebel
_et al._ (2021)) can then be regarded as an effective squeezed reservoir of
the mechanical mode. The squeezing parameter and the reference phase of the
auxiliary reservoir could be controlled by the amplitude and phase of the pump
tone used to modulate the magnetic flux through the SQUID.

Appendix C System parameters
Table 1 shows the symbols, names and values of the main system parameters that
are used in our numerical calculations.

Appendix D Derivation of the effective coupling between the magnon and the NV
spin

To begin with, we transform the Hamiltonian (20) in the main text into the
interaction picture, giving

$\begin{split}\frac{\hat{H}_{\mathrm{Hybrid}}^{\mathrm{S,I}}}{\hbar}=&g_{r}\hat{b}_{s}\hat{s}^{\dagger}e^{\mathrm{-i}(\Delta_{s}-\Delta_{K})t}\\!+\\!g_{r}\hat{b}_{s}^{\dagger}\hat{s}^{\dagger}e^{\mathrm{i}(\Delta_{s}+\Delta_{K}+2\omega_{p})t}\\\
&\\!-\\!g_{c}\hat{b}_{s}^{\dagger}\hat{s}^{\dagger}e^{\mathrm{i}(\Delta_{s}+\Delta_{K})t}\\!-\\!g_{c}\hat{b}_{s}\hat{s}^{\dagger}e^{\mathrm{-i}(\Delta_{s}-\Delta_{K}-2\omega_{p})t}\\\
&\\!+\\!\lambda_{r}\hat{b}_{s}\hat{\sigma}_{+}e^{\mathrm{-i}(\Delta_{s}-\Delta_{\mathrm{NV}})t}\\!+\\!\lambda_{r}\hat{b}_{s}^{\dagger}\hat{\sigma}_{+}e^{\mathrm{-i}(\Delta_{s}+\Delta_{\mathrm{NV}}+2\omega_{p})t}\\\
&\\!-\\!\lambda_{c}\hat{b}_{s}^{\dagger}\hat{\sigma}_{+}e^{\mathrm{i}(\Delta_{s}+\Delta_{\mathrm{NV}})t}\\!-\\!\lambda_{c}\hat{b}_{s}\hat{\sigma}_{+}e^{\mathrm{-i}(\Delta_{s}-\Delta_{\mathrm{NV}}-2\omega_{p})t}\\\
&+\mathrm{H.c.},\end{split}$ (33)

In order to eliminate the undesired terms that oscillate rapidly in the
Hamiltonian above and simultaneously eliminate the mechanical degree of
freedom, we assume the following large detuning conditions

$\begin{split}\min\big{\\{}&\left|\Delta_{s}+\Delta_{K}\right|,\left|\Delta_{s}+\Delta_{K}+2\omega_{p}\right|,\\\
&\left|\Delta_{s}-\Delta_{K}-2\omega_{p}\right|\big{\\}}\gg\left|\Delta_{s}-\Delta_{K}\right|\gg
g_{r},\\\
\min\big{\\{}&\left|\Delta_{s}+\Delta_{\mathrm{NV}}\right|,\left|\Delta_{s}+\Delta_{\mathrm{NV}}+2\omega_{p}\right|,\\\
&\left|\Delta_{s}-\Delta_{\mathrm{NV}}-2\omega_{p}\right|\big{\\}}\gg\left|\Delta_{s}-\Delta_{\mathrm{NV}}\right|\gg\lambda_{r}.\end{split}$
(34)

In this case, the mode $\hat{b}_{s}$ is only virtually excited and can then be
eliminated adiabatically. Moreover, we assume that the phonon mode is
initially in the vacuum state, and can always remain in the vacuum due to the
effective decoupling from the system dynamics. Given above, we obtain the
effective Hamiltonian involving only the magnon and spin degrees of freedom

$\begin{split}\frac{\hat{H}_{\mathrm{ms}}^{\mathrm{S}}}{\hbar}=&\frac{g_{r}^{2}}{\Delta_{K}-\Delta_{s}}\hat{s}^{\dagger}\hat{s}+\frac{\lambda_{r}^{2}}{\Delta_{\mathrm{NV}}-\Delta_{s}}\hat{\sigma}_{+}\hat{\sigma}_{-}\\\
&+\frac{g_{r}\lambda_{r}}{2}(\frac{1}{\Delta_{K}-\Delta_{s}}+\frac{1}{\Delta_{\mathrm{NV}}-\Delta_{s}})(\hat{s}^{\dagger}\hat{\sigma}_{-}+\hat{s}\hat{\sigma}_{+}).\end{split}$
(35)

We recall the realistic parameters considered in the main text,
$\omega_{x}\approx 2\pi\times 3.8$ MHz,
$\omega_{K}=\omega_{\mathrm{NV}}\approx 2\pi\times 0.96$ GHz, and
$g=\lambda\approx 2\pi\times 0.69$ MHz. Then, the Hamiltonian above can be
further reduced, yielding the Hamiltonian (21) in the main text. Moreover,
considering the MPD with, e.g., $r_{p}=1.54$ and $\Omega_{p}\approx 2\pi\times
1.05$ GHz, we have $\Delta_{s}\approx-55g_{r}$ and
$\Delta_{K}\approx-45g_{r}$, which validates the large detuning conditions
(34) and therefore the effective Hamiltonian (35).