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# Entangling ferrimagnetic magnons with an atomic ensemble via opto-
magnomechanics

Zhi-Yuan Fan  Interdisciplinary Center of Quantum Information, State Key
Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key
Laboratory of Quantum Technology and Device, School of Physics, Zhejiang
University, Hangzhou 310027, China  Hang Qian  Interdisciplinary Center of
Quantum Information, State Key Laboratory of Modern Optical Instrumentation,
and Zhejiang Province Key Laboratory of Quantum Technology and Device, School
of Physics, Zhejiang University, Hangzhou 310027, China  Xuan Zuo
Interdisciplinary Center of Quantum Information, State Key Laboratory of
Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of
Quantum Technology and Device, School of Physics, Zhejiang University,
Hangzhou 310027, China  Jie Li  Interdisciplinary Center of Quantum
Information, State Key Laboratory of Modern Optical Instrumentation, and
Zhejiang Province Key Laboratory of Quantum Technology and Device, School of
Physics, Zhejiang University, Hangzhou 310027, China

###### Abstract

We show how to prepare macroscopic entanglement between an atomic ensemble and
a large number of magnons in a ferrimagnetic YIG crystal. Specifically, we
adopt an opto-magnomechanical configuration where the magnetostriction-induced
magnomechanical displacement couples to an optical cavity via radiation
pressure, and the latter further couples to an ensemble of two-level atoms
that are placed inside the cavity. We show that by properly driving the cavity
and magnon modes, optomechanical entanglement is created which is further
distributed to the atomic and magnonic systems, yielding stationary
entanglement between atoms and magnons. The atom-magnon entanglement is a
result of the combined effect of opto- and magnomechanical cooling and
optomechanical parametric down-conversion interactions. A competition
mechanism between two mechanical cooling channels is revealed. Our results
indicate that the hybrid opto-magnomechanical system may become a promising
system for preparing macroscopic quantum states involving magnons, photons,
phonons and atoms.

Cavity optomechanics (COM) explores the interaction between the
electromagnetic field and mechanical motion via radiation pressure RMP2014 .
The past decade has witnessed significant progress in the field of COM in
experimentally preparing macroscopic quantum states of massive mechanical
oscillators, including the realization of entanglement between a mechanical
oscillator and an electromagnetic field Palomaki , entanglement between two
mechanical oscillators mechanical1 ; mechanical2 ; mechanical3 , and quantum
squeezing of mechanical motion qusqz , etc.

In analogy to cavity optomechanics, cavity magnomechanics (CMM) 16SciAdv ;
JieLi18 ; PRX21 ; RCShen2022  has recently received increasing attention
because of its potential for preparing quantum states at larger scales JieLi18
; JL19PRA ; JL19NJP ; Tan19PRre ; Ding ; JLQST21 ; OE21 ; ChenPRA21 ; JLNSR22
; HussainPRA22 ; QiuPRA22 ; HQianQST23 ; Asjad23 , as well as its various
promising applications in quantum information science and quantum technologies
SQapp1 ; SQapp2 ; SQapp3 ; Ding19 ; JieLi20 ; SQapp5 ; SQapp6 ; SQapp7 ;
SQapp8 ; FanPRA21 ; SQapp10 ; SQapp11 . It studies interactions between
microwave cavity photons, magnons (quanta of the spin wave), and
magnetostriction-induced vibration phonons in magnetically ordered materials,
such as yttrium-iron-garnet (YIG) 16SciAdv ; JieLi18 ; PRX21 ; RCShen2022 .
The combination of COM and CMM, realized by coupling the magnomechanical
displacement to an optical cavity via radiation pressure, forms the new system
of opto-magnomechanics (OMM) FanPRA21 ; QST23 ; Fanarxiv . Such a hybrid
system enables us to optically read out magnon population FanPRA21  and
arbitrary magnonic quantum states in solids QST23 , and prepare optomagnonic
QST23  and microwave-optics Fanarxiv  entanglement. Thus, the system would
find promising applications in quantum information processing and quantum
networks.

In this article, we show how to prepare macroscopic entanglement between
ferrimagnetic magnons and an atomic ensemble by using such a novel OMM system.
Specifically, we further couple the optical cavity of the OMM to an ensemble
of two-level atoms that are initially prepared in their ground state. Keeping
the atoms and magnons in the low-excitation limit, which allows us to bosonize
the atomic polarization and the collective spins, the system then becomes a
four-mode bosonic system. By strongly driving the cavity with a red-detuned
laser, the mechanical motion scatters the driving photons onto two sidebands.
When the cavity resonance and atomic frequencies match the anti-Stokes and
Stokes sidebands, respectively, an entangled state between atoms and vibration
phonons can be created. By further driving the magnons with a relatively weak
red-detuned microwave field, which activates the magnon-phonon state-swap
interaction, the atom-phonon entanglement is then partially distributed to
magnons, yielding a macroscopic entangled state of magnons and atoms. The
entangled state is stationary and robust against bath temperature. A strategy
is provided to detect the entanglement.

Figure 1: (a)-(b) Sketch of the atom-OMM system. An optical cavity mode ($c$)
driven by a laser at frequency $\omega_{L}$ couples to an ensemble of two-
level atoms ($a$) and a magnon mode ($m$) in a YIG crystal by the mediation of
a mechanical vibration mode ($b$) induced by magnetostriction. (c) Mode
frequenies and linewidths adopted in the protocol. In the resolved sideband
limit ($\omega_{b}\gg\kappa_{c,m}$), when the cavity and atomic frequencies
match the anti-Stokes sideband at $\omega_{L}+\omega_{b}$ and the Stokes
sideband at $\omega_{L}-\omega_{b}$ of the laser drive field, respectively,
and when further the magnon mode is resonant with the anti-Stokes sideband at
$\omega_{0}+\omega_{b}$ of the microwave drive field, stationary entanglement
between atoms and magnons is established.

The system we consider is depicted in Fig. 1(a), which consists of optical
cavity photons, magnons in a YIG crystal, magnetostriction-induced vibration
phonons, and an ensemble of two-level atoms. The magnomechanical displacement
couples to the optical cavity via radiation pressure (a dispersive
interaction), e.g., by attaching a small highly reflective mirror pad onto the
surface of a YIG micro bridge QST23 ; bridge ; Fanarxiv . The YIG bridge is
micron-sized and can support long-lived spin-wave excitations with the
frequency in gigahertz and mechanical vibration phonons with the frequency
ranging from tens to hundreds of megahertz bridge . The large frequency
mismatch of the magnon and mechanical modes permits a dominant magnon-phonon
dispersive coupling QST23 . Note that the attached mirror pad should be
fabricated sufficiently small compared to the YIG bridge, such that there is
negligible bending displacement (relative motion), and the YIG bridge and the
attached mirror can stick together tightly, which oscillate approximately with
the same frequency. Alternatively, one may adopt the ‘membrane-in-the-middle’
configuration Jack  by placing the YIG bridge in the middle of the optical
cavity, which can also realize the dispersive coupling between the
magnomechanical displacement and the optical cavity.

The Hamiltonian of the system is given by

$\begin{split}H/\hbar=&\
\omega_{c}c^{\dagger}c+\omega_{m}m^{\dagger}m+\frac{\omega_{a}}{2}S_{z}+\frac{\omega_{b}}{2}\left(q^{2}+p^{2}\right)\\\
&+g_{a}\left(S_{+}c+S_{-}c^{\dagger}\right)-g_{c}c^{\dagger}cq+g_{m}m^{\dagger}mq+H_{\mathrm{dri}}/\hbar,\end{split}$
(1)

where $c$ ($c^{\dagger}$) and $m$ ($m^{\dagger}$) are the annihilation
(creation) operators of the cavity and magnon modes, respectively, satisfying
$[k,k^{{\dagger}}]=1$ ($k=c,m$). $q$ and $p$ ($[q,p]=i$) are the dimensionless
position and momentum of the mechanical mode. The collective spin operators of
an ensemble of $N_{a}$ two-level atoms with natural frequency $\omega_{a}$,
$S_{\pm,z}=\Sigma_{i=1}^{N_{a}}\sigma^{(i)}_{\pm,z}$, with $\sigma_{\pm,z}$
being the Pauli matrices, which satisfy the commutation relations
$[S_{+},S_{-}]=S_{z}$ and $[S_{z},S_{\pm}]=\pm 2S_{\pm}$. $\omega_{c}$,
$\omega_{m}$ and $\omega_{b}$ are the resonance frequencies of the cavity,
magnon and mechanical modes, respectively, and the magnon frequency can be
adjusted in a large range by varying the strength of the bias magnetic field
$B_{0}$. The atom-cavity coupling strength
$g_{a}=\mu\sqrt{\omega_{c}/2\hbar\epsilon_{0}V_{c}}$, with $\mu$ being the
atomic dipole moment, $V_{c}$ the cavity mode volume, and $\epsilon_{0}$ the
vacuum permittivity. $g_{c}$ ($g_{m}$) denotes the bare optomechanical
(magnomechanical) coupling strength, which can be greatly enhanced by strongly
driving the cavity (magnon) mode. The last term is the driving Hamiltonian,
$H_{\mathrm{dri}}=i\hbar
E(c^{\dagger}e^{-i\omega_{L}t}-\mathrm{H.c.})+i\hbar\Omega_{d}(m^{\dagger}e^{-i\omega_{0}t}-\mathrm{H.c.})$,
where $E=\sqrt{2\kappa_{c}P_{L}/\hbar\omega_{L}}$ represents the coupling
strength between the cavity and the laser drive field, with $P_{L}$
($\omega_{L}$) being the power (frequency) of the laser, and $\kappa_{c}$ the
cavity decay rate. The Rabi frequency
$\Omega_{d}=\frac{\sqrt{5}}{4}\gamma\sqrt{N}B_{d}$ JieLi18  denotes the
coupling between the magnon mode and the drive magnetic field with amplitude
$B_{d}$ and frequency $\omega_{0}$, and $\gamma$ is the gyromagnetic ratio and
$N$ is the number of spins in the YIG crystal. Note that the magnons are
assumed in the low-excitation limit, $\langle m^{\dagger}m\rangle\ll 2Ns$,
where $s=\frac{5}{2}$ is the spin number of the ground state ${\rm Fe}^{\rm
3+}$ ion in YIG JieLi18 . This validates the bosonic description of the spins,
where the system can be well described by a harmonic oscillator SQapp1 .

The dynamics of the system governed by the Hamiltonian (1) is generally
complicated. It, however, can be simplified by assuming the atoms in the low-
excitation limit, where the excitation probability of a single atom is small
DV08 . We assume that the atoms are initially prepared in their ground state,
so that $S_{z}\simeq\langle S_{z}\rangle\simeq-N_{a}$, and are off-resonantly
coupled to the optical cavity. In this case, the dynamics of the atomic
polarization can be described by bosonic operators. Specifically, the atomic
annihilation operator can be defined as $a=S_{-}/\sqrt{|\langle
S_{z}\rangle|}$, and it satisfies the bosonic commutation relation
$[a,a^{{\dagger}}]=1$. Consequently, we obtain the fully bosonized
Hamiltonian, given by

$\begin{split}H/\hbar=&\sum_{j=a,c,m}\omega_{j}j^{\dagger}j+\frac{\omega_{b}}{2}\left(q^{2}+p^{2}\right)+H_{\mathrm{dri}}/\hbar\\\
&+g_{N}\left(a^{\dagger}c+ac^{\dagger}\right)-g_{c}c^{\dagger}cq+g_{m}m^{\dagger}mq,\end{split}$
(2)

where $g_{N}=g_{a}\sqrt{N_{a}}$ is the effective atom-cavity coupling
strength.

By including dissipation and input noise of each mode and working in the
interaction picture with respect to
$\hbar\omega_{L}(a^{\dagger}a+c^{\dagger}c)+\hbar\omega_{0}m^{\dagger}m$, we
obtain the following quantum Langevin equations (QLEs) of the system:

$\begin{split}\dot{a}=&-i\Delta_{a}a-\gamma_{a}a-ig_{N}c+\sqrt{2\gamma_{a}}a_{\rm
in},\\\ \dot{c}=&-i\Delta_{c}c-\kappa_{c}c+ig_{c}cq-
ig_{N}a+E+\sqrt{2\kappa_{c}}c_{\rm in},\\\
\dot{m}=&-i\Delta_{m}m-\kappa_{m}m-ig_{m}mq+\Omega_{d}+\sqrt{2\kappa_{m}}m_{\rm
in},\\\ \dot{q}=&\
\omega_{b}p,\,\,\,\,\,\dot{p}=-\omega_{b}q-\gamma_{b}p+g_{c}c^{\dagger}c-g_{m}m^{\dagger}m+\xi,\end{split}$
(3)

where $\Delta_{a(c)}=\omega_{a(c)}-\omega_{L}$ and
$\Delta_{m}=\omega_{m}-\omega_{0}$. $\gamma_{a}$ is the decay rate of the
atomic excited level and $\gamma_{b}$ ($\kappa_{m}$) is the dissipation rate
of the mechanical (magnon) mode. $j_{\rm in}(t)$ ($j=a,c,m$) denote the zero-
mean input noise operators, which obey the following correlation functions
$\langle j_{\rm in}^{\dagger}(t)j_{\rm
in}(t^{\prime})\rangle=N_{j}(\omega_{j})\delta(t-t^{\prime})$ and $\langle
j_{\rm in}(t)j^{\dagger}_{\rm
in}(t^{\prime})\rangle=[N_{j}(\omega_{j})+1]\delta(t-t^{\prime})$. $\xi(t)$ is
the Hermitian Brownian noise operator acting on the mechanical oscillator,
which is intrinsically non-Markovian, but a Markovian approximation can be
taken for a large mechanical quality factor $Q_{b}=\omega_{b}/\gamma_{b}\gg 1$
DV01 . In this case, $\xi(t)$ takes a $\delta$-autocorrelation:
$\langle\xi(t)\xi(t^{\prime})+\xi(t^{\prime})\xi(t)\rangle/2\simeq\gamma_{b}[2N_{b}(\omega_{b})+1]\delta(t-t^{\prime})$.
Here, $N_{k}(\omega_{k})=[\mathrm{exp}(\hbar\omega_{k}/k_{B}T)-1]^{-1}$
($k=a,c,m,b$) are the mean thermal excitation number of each mode at bath
temperature $T$, with $k_{B}$ as the Boltzmann constant.

The creation of strong quantum correlations, like entanglement, in the system
requires sufficiently strong opto- and magnomechanical interactions. To this
end, we drive the cavity (magnon) mode with a strong laser (microwave) field,
which leads to large steady-state amplitudes $|\langle c\rangle|$, $|\langle
m\rangle|\gg 1$. This allows us to linearize the nonlinear opto- and
magnomechanical dynamics around the steady state, which is implemented by
writing each mode operator as the sum of its classical average and quantum
fluctuation operator, i.e., $k=\langle k\rangle+\delta k$, and neglecting
small second-order fluctuation terms. We aim to study quantum correlations
among the macroscopic subsystems, and thus we focus on the dynamics of the
quantum fluctuations (around the steady-state averages). The linearized QLEs
describing the quantum fluctuations $(\delta x_{a},\delta y_{a},\delta
x_{c},\delta y_{c},\delta q,\delta p,\delta x_{m},\delta y_{m})$, where
$\delta x_{j}=(\delta j+\delta j^{\dagger})/\sqrt{2}$ and $\delta
y_{j}=i(\delta j^{\dagger}-\delta j)/\sqrt{2}$ ($j=a,c,m$) denote the
fluctuations of the amplitude and phase quadratures of the corresponding mode,
can be written in the matrix form of

$\dot{u}(t)=Au(t)+n(t),$ (4)

where $u=\left(\delta x_{a},\delta y_{a},\delta x_{c},\delta y_{c},\delta
q,\delta p,\delta x_{m},\delta y_{m}\right)^{\rm T}$ and
$n=\big{(}\sqrt{2\gamma_{a}}x_{a}^{\rm in},\sqrt{2\gamma_{a}}y_{a}^{\rm
in},\sqrt{2\kappa_{c}}x_{c}^{\rm in},\sqrt{2\kappa_{c}}y_{c}^{\rm
in},0,\xi,\sqrt{2\kappa_{m}}x_{m}^{\rm in},\sqrt{2\kappa_{m}}y_{m}^{\rm
in}\big{)}^{\rm T}$, and the drift matrix $A$ is given by

$A=\begin{pmatrix}-\gamma_{a}&\Delta_{a}&0&g_{N}&0&0&0&0\\\
-\Delta_{a}&-\gamma_{a}&-g_{N}&0&0&0&0&0\\\
0&g_{N}&-\kappa_{c}&\tilde{\Delta}_{c}&G_{c}&0&0&0\\\
-g_{N}&0&-\tilde{\Delta}_{c}&-\kappa_{c}&0&0&0&0\\\
0&0&0&0&0&\omega_{b}&0&0\\\ 0&0&0&-G_{c}&-\omega_{b}&-\gamma_{b}&0&G_{m}\\\
0&0&0&0&-G_{m}&0&-\kappa_{m}&\tilde{\Delta}_{m}\\\
0&0&0&0&0&0&-\tilde{\Delta}_{m}&-\kappa_{m}\end{pmatrix},$ (5)

where the effective detunings $\tilde{\Delta}_{c}=\Delta_{c}-g_{c}\langle
q\rangle$ and $\tilde{\Delta}_{m}=\Delta_{m}+g_{m}\langle q\rangle$, which
include the frequency shifts due to the mechanical displacement $\langle
q\rangle=\big{(}g_{c}|\langle c\rangle|^{2}-g_{m}|\langle
m\rangle|^{2}\big{)}/\omega_{b}$, jointly caused by the opto- and
magnomechanical interactions. The effective opto- and magnomechanical coupling
strengths are $G_{c}=i\sqrt{2}g_{c}\langle c\rangle$ and
$G_{m}=i\sqrt{2}g_{m}\langle m\rangle$, which are significantly enhanced by
the large amplitudes of the cavity and magnon modes:

$\begin{split}\langle
m\rangle=\frac{\Omega_{d}}{\kappa_{m}+i\tilde{\Delta}_{m}},\,\,\,\,\,\,\langle
c\rangle=\frac{E(\gamma_{a}+i\Delta_{a})}{g_{N}^{2}+(\gamma_{a}+i\Delta_{a})(\kappa_{c}+i\tilde{\Delta}_{c})}.\end{split}$
(6)

The amplitude of the atomic mode can be obtained by $\langle
a\rangle=-ig_{N}\langle c\rangle/(\gamma_{a}+i\Delta_{a})$. It should be noted
that the drift matrix $A$ in Eq.(5) is derived under the condition that
$|\Delta_{a}|,|\tilde{\Delta}_{c}|,|\tilde{\Delta}_{m}|\gg\gamma_{a},\kappa_{c},\kappa_{m}$.
This yields simpler approximate expressions $\langle
m\rangle\simeq-i\Omega_{d}/\tilde{\Delta}_{m}$ and $\langle c\rangle\simeq
iE\Delta_{a}/(g_{N}^{2}-\Delta_{a}\tilde{\Delta}_{c})$, which are pure
imaginary numbers and thus give rise to approximately real couplings $G_{m}$
and $G_{c}$. In fact, as will be shown later, the condition
$|\Delta_{a}|,|\tilde{\Delta}_{c}|,|\tilde{\Delta}_{m}|\simeq\omega_{b}\gg\gamma_{a},\kappa_{c},\kappa_{m}$
(c.f. Fig. 1(c)) corresponds to the resolved sideband limit, and is optimal to
generate entanglement in the system JieLi18 ; JL19NJP ; JieLi20 ; QST23 .

Since the dynamics of the system is fully linearized and the input noises are
Gaussian, the state of the system at any given time is Gaussian, which can be
characterized by an $8\times 8$ covariance matrix (CM) $\mathcal{V}$, with its
entries defined as $\mathcal{V}_{ij}=\langle
u_{i}(t)u_{j}(t^{\prime})+u_{j}(t^{\prime})u_{i}(t)\rangle/2$
($i,j=1,2,...,8$). To get the steady-state CM $\mathcal{V}$, one can directly
solve the Lyapunov equation

$A\mathcal{V}+\mathcal{V}A^{T}=-D,$ (7)

where the diffusion matrix
$D=\mathrm{diag}[\gamma_{a},\gamma_{a},\kappa_{c},\kappa_{c},0,\gamma_{b}(2N_{b}+1),\kappa_{m}(2N_{m}+1),\kappa_{m}(2N_{m}+1)]$,
which is defined by $D_{ij}\delta(t-t^{\prime})=\langle
n_{i}(t)n_{j}(t^{\prime})+n_{j}(t^{\prime})n_{i}(t)\rangle/2$. In obtaining
$D$, we take $N_{a(c)}\simeq 0$ due to their high mode frequencies. When the
CM $\mathcal{V}$ is achieved, we can then quantify the entanglement between
any two subsystems using the logarithmic negativity $E_{N}$ adesso , defined
as

$E_{N}=\textup{max}\left[0,-\textup{ln}(2\eta^{-})\right],$ (8)

where $\eta^{-}\equiv
2^{-1/2}\,\big{[}\Sigma-\big{(}\Sigma^{2}-4\,\textup{det}\,{\cal
V}_{4}\big{)}^{1/2}\big{]}^{1/2}$, and ${\cal V}_{4}=\big{[}{\cal V}_{e},{\cal
V}_{ef};{\cal V}_{ef}^{\rm T},{\cal V}_{f}\big{]}$ is the $4\times 4$ CM of
the bipartite system involving mode $e$ and $f$ ($e,f=a,c,m,b$), with ${\cal
V}_{e},{\cal V}_{f}$ and ${\cal V}_{ef}$ being the $2\times 2$ blocks of
${\cal V}_{4}$, and $\Sigma\equiv\textup{det}\,{\cal
V}_{e}+\textup{det}\,{\cal V}_{f}-2\textup{det}\,{\cal V}_{ef}$.

Figure 2: Density plot of steady-state (a) photon-phonon entanglement
$E_{cb}$, (b) atom-phonon entanglement $E_{ab}$, and (c) atom-magnon
entanglement $E_{am}$ versus detunings $\Delta_{a}$ and $\tilde{\Delta}_{c}$
(in units of $\omega_{b}$). (d) Atom-magnon entanglement $E_{am}$ versus
$\Delta_{a}$ and magnomechanical coupling strength $G_{m}$ (in units of
$G_{c}$). We take $G_{c}/2\pi=8$ MHz and $\tilde{\Delta}_{m}=\omega_{b}$ in
all plots, $\tilde{\Delta}_{c}=0.5\omega_{b}$ in (d), and $G_{m}/2\pi=2.5$ MHz
in (a)-(c). See text for the other parameters.

The mechanical mode has the lowest resonance frequency in the system,
typically in megahertz bridge ; 16SciAdv ; JieLi18 ; PRX21 ; RCShen2022 , and
is highly thermally populated even at cryogenic temperatures. Therefore,
cooling the mechanical mode close to the ground state is a requisite for
preparing quantum states in the system JieLi18 . To achieve this, we drive the
cavity with a red-detuned laser, which activates the optomechanical anti-
Stokes scattering and can significantly cool the mechanical mode in the
resolved sideband limit $\omega_{b}\gg\kappa_{c}$ RMP2014 . We use a
relatively strong laser power, which yields a strong optomechanical coupling
$G_{c}$ and breaks the weak-coupling condition $G_{c}\ll\omega_{b}$ for taking
the rotating-wave (RW) approximation to obtain the optimal beam-splitter
interaction $\propto c^{\dagger}b+cb^{\dagger}$
($b\,{=}\,\frac{q+ip}{\sqrt{2}}$) for cooling. The counter-RW terms $\propto
c^{\dagger}b^{\dagger}+cb$, corresponding to the parametric down-conversion
(PDC) interaction, then start to play the role and generate the optomechanical
entanglement DV07PRL , as shown in Fig. 2(a). The anti-crossing around
$\tilde{\Delta}_{c}=\omega_{b}$ in the figure is a signature of the strong
coupling. The entanglement can be distributed to the atomic system when the
atomic frequency matches the Stokes sideband, i.e., $\Delta_{a}=-\omega_{b}$
DV08  (c.f. Fig. 1(c)), giving rise to the atom-phonon entanglement, as
confirmed by Fig. 2(b). A similar mechanism has been adopted to prepare
entangled states in CMM systems JieLi18 ; JL19NJP ; JieLi20 . By further
driving the magnon mode with a red-detuned microwave field with detuning
$\tilde{\Delta}_{m}=\omega_{b}$, the magnomechanical anti-Stokes scattering is
activated, which realizes the magnon-phonon state-swap operation. As a result,
the atom-phonon entanglement is further distributed to the magnonic system,
yielding stationary atom-magnon entanglement, as seen in Fig. 2(c). This
entanglement transfer becomes efficient only when the magnomechanical coupling
is sufficiently strong. This is clearly shown in Fig. 2(d) that the atom-
magnon entanglement $E_{am}$ increases as the grow of the coupling $G_{m}$,
which increases the magnon-phonon state-swap efficiency.

In view of the whole process, from the perspective of entanglement as a finite
quantum resource, it is originally generated in the optomechanical system, and
then distributed to the atom-phonon system via the cavity-atom linear
coupling, and further to the atom-magnon system via the phonon-magnon state-
swap interaction. The complementary distributions of the entanglement in Figs.
2(a), 2(b) and 2(c) are a clear sign of such an entanglement transfer process.
This feature was also observed in other multipartite systems JieLi18 ;
Fanarxiv .

In plotting Fig. 2, we have used feasible parameters bridge ; 16SciAdv ;
JieLi18 ; PRX21 ; RCShen2022 ; QST23 : $\omega_{m}/2\pi=10$ GHz,
$\omega_{b}/2\pi=40$ MHz, $\lambda_{L}=1064$ nm (optical wavelength),
$\kappa_{m}/2\pi=1$ MHz, $\gamma_{a}=\kappa_{m}$, $\kappa_{c}=2\kappa_{m}$,
$\gamma_{b}/2\pi=10^{2}$ Hz, $G_{c}/2\pi=8$ MHz, $g_{N}/2\pi=8$ MHz,
$G_{m}/2\pi=2.5$ MHz, $\tilde{\Delta}_{m}=\omega_{b}$, and $T=10$ mK. The
strong optomechanical coupling $G_{c}/2\pi=8$ MHz can be achieved with a laser
power $P_{L}\simeq 4.4$ mW for $g_{c}/2\pi=1$ kHz at detunings
$\tilde{\Delta}_{c}\simeq 0.5\omega_{b}$ and $\Delta_{a}=-\omega_{b}$. The
magnomechanical coupling $G_{m}/2\pi=2.5$ MHz corresponds to a microwave drive
power $P_{0}\simeq 1.44$ mW for a $5\times 2\times 1$ $\mu m^{3}$ YIG bridge
(approximated as a cuboid) with $g_{m}/2\pi=20$ Hz QST23 .

Figure 3: (a) Stationary atom-magnon entanglement $E_{am}$ (dashed) and the
corresponding optimal cavity detuning $\tilde{\Delta}_{c}^{\rm{opt}}$ (solid,
in unit of $\omega_{b}$ ) versus $G_{m}$. (b) $E_{am}$ versus bath temperature
$T$. We take $\tilde{\Delta}_{m}=-\Delta_{a}=\omega_{b}$ in both plots, and
$\tilde{\Delta}_{c}=0.5\omega_{b}$ and $G_{m}/2\pi=2.5$ MHz in (b). The other
parameters are the same as in Fig. 2(c).

In our system, since the cavity and magnon modes are simultaneously driven by
red-detuned laser and microwave fields (Fig. 1(c)), both the opto- and
magnomechanical anti-Stokes scatterings are present in the system and these
two mechanical cooling channels may compete when the drive fields are
sufficiently strong. Such competition effect is revealed in Fig. 3(a), where
we plot the atom-magnon entanglement $E_{am}$ and the corresponding optimal
cavity-laser detuning $\tilde{\Delta}_{c}^{\rm{opt}}$ versus the
magnomechanical coupling $G_{m}$. When $G_{m}$ is small, the optomechanical
cooling channel is dominant and thus the optimal detuning
$\tilde{\Delta}_{c}^{\rm{opt}}$ (for the entanglement) is around $\omega_{b}$,
at which the cooling efficiency is maximized RMP2014 . As $G_{m}$ grows, the
magnon-phonon state-swap efficiency increases yielding an increasing $E_{am}$
(dashed line). Besides, the contribution of the magnomechanical anti-Stokes
scattering in mechanical cooling increases, which leads to a decreasing
optimal detuning $\tilde{\Delta}_{c}^{\rm{opt}}\to 0.1\omega_{b}$ (solid
line). Such a shift of the optimal detuning implies that the strength of the
optomechanical beam-splitter interaction reduces (its role in mechanical
cooling is gradually replaced by the magnomechanical beam-splitter
interaction), while the strength of the optomechanical PDC interaction
increases. The optimal detuning $\tilde{\Delta}_{c}^{\rm{opt}}$ is the result
of the trade-off among the magnomechanical cooling, the optomechanical cooling
and PDC. It is worth noting that the steep drop of
$\tilde{\Delta}_{c}^{\rm{opt}}$ as $G_{m}$ grows in Fig. 3(a) is due to the
anti-crossing associated with the strong coupling (c.f. Fig. 2), corresponding
to the sudden transition of $\tilde{\Delta}_{c}^{\rm{opt}}$ between two
branches near the anti-crossing. The generated stationary macroscopic atom-
magnon entanglement can exist for bath temperature up to hundreds of
millikelvin, as show in Fig. 3(b), under a moderate coupling $G_{m}/2\pi=2.5$
MHz.

Lastly, we discuss the validity of our model and provide a strategy to detect
and verify the entanglement. The bosonic description for the spin systems is
valid in the low-excitation limit, i.e., $\langle m^{\dagger}m\rangle\ll 2Ns$
and $\langle a^{\dagger}a\rangle\ll N_{a}$. Under the parameters of Fig. 3(b),
we obtain $\langle m^{\dagger}m\rangle=7.7\times 10^{9}\ll 2Ns=2.1\times
10^{11}$ and $\langle a^{\dagger}a\rangle=1.3\times 10^{6}\ll N_{a}=6.4\times
10^{7}$. In estimating $N_{a}$, we take $g_{a}/2\pi=10^{3}$ Hz DV08 . Clearly,
the low-excitation condition is well fulfilled. For the entanglement
detection, the magnon state can be accessed by coupling to a microwave cavity
that is driven by a weak probe field. Due to the magnon-cavity beam-splitter
coupling naka ; zou , the magnon state can be read out in the cavity output
field. By homodyning the microwave output field, two quadratures of the magnon
mode can be measured. Similarly, the atomic polarization quadratures can be
measured by coupling to an additional optical cavity that is driven by a weak
laser field. Having measured the magnonic and atomic quadratures, one can then
build the CM, based on which the atom-magnon entanglement is computed Palomaki
; mechanical3 ; DV07PRL .

In conclusion, we present a protocol to prepare stationary entangled states of
two macroscopic systems, an atomic ensemble and ferrimagnetic magnons, in an
opto-magnomechanical system. The atom-magnon entanglement is established as a
result of the combination of opto- and magnomechanical cooling and
optomechanical PDC interactions, and the entanglement distribution among
different subsystems. A competition mechanism between two opto- and
magnomechanical cooling channels is revealed. The protocol may find potential
applications in the macroscopic quantum studies and quantum information
processing with hybrid systems hb1 ; hb2 ; hb3 .

This work has been supported by National Key Research and Development Program
of China (Grant No. 2022YFA1405200) and National Natural Science Foundation of
China (No. 92265202).

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