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# Quantum network with magnonic and mechanical nodes

Jie Li  Interdisciplinary Center of Quantum Information, State Key Laboratory
of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of
Quantum Technology and Device, Department of Physics, Zhejiang University,
Hangzhou 310027, China  Yi-Pu Wang  Interdisciplinary Center of Quantum
Information, State Key Laboratory of Modern Optical Instrumentation, and
Zhejiang Province Key Laboratory of Quantum Technology and Device, Department
of Physics, Zhejiang University, Hangzhou 310027, China  Wei-Jiang Wu
Interdisciplinary Center of Quantum Information, State Key Laboratory of
Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of
Quantum Technology and Device, Department of Physics, Zhejiang University,
Hangzhou 310027, China  Shi-Yao Zhu  Interdisciplinary Center of Quantum
Information, State Key Laboratory of Modern Optical Instrumentation, and
Zhejiang Province Key Laboratory of Quantum Technology and Device, Department
of Physics, Zhejiang University, Hangzhou 310027, China  J. Q. You
Interdisciplinary Center of Quantum Information, State Key Laboratory of
Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of
Quantum Technology and Device, Department of Physics, Zhejiang University,
Hangzhou 310027, China

###### Abstract

A quantum network consisting of magnonic and mechanical nodes connected by
light is proposed. Recent years have witnessed a significant development in
cavity magnonics based on collective spin excitations in ferrimagnetic
crystals, such as yttrium iron garnet (YIG). Magnonic systems are considered
to be a promising building block for a future quantum network. However, a
major limitation of the system is that the coherence time of the magnon
excitations is limited by their intrinsic loss (typically in the order of 1
$\mu$s for YIG). Here, we show that by coupling the magnonic system to a
mechanical system using optical pulses, an arbitrary magnonic state (either
classical or quantum) can be transferred to and stored in a distant long-lived
mechanical resonator. The fidelity depends on the pulse parameters and the
transmission loss. We further show that the magnonic and mechanical nodes can
be prepared in a macroscopic entangled state. These demonstrate the quantum
state transfer and entanglement distribution in such a novel quantum network
of magnonic and mechanical nodes. Our work shows the possibility to connect
two separate fields of optomagnonics and optomechanics, and to build a long-
distance quantum network based on magnonic and mechanical systems.

##  I Introduction

Hybrid quantum systems, composed of distinct physical systems with
complementary functionalities, provide diverse novel platforms and promising
opportunities for applications in quantum technologies, quantum-information
processing and quantum sensing hybrid0 ; hybrid1 ; hybrid2 . It merits our
particular attention that, during the past decade a rapid and significant
progress has been made in the field of cavity magnonics, based on coherently
coupled microwave cavity photons and collective spin excitations in the
ferrimagnetic material of yttrium iron garnet (YIG) NakaRev ; NatRev ; RMP ;
S1 ; S2 ; S3 ; S5 ; S6 ; S7 ; S8 ; S9 ; S9b ; S10 ; S11 ; S12 ; S13 ; S14 ;
S15 ; S16 ; S17 ; S18 ; S19 ; S20 ; S21 ; S22 ; S23 ; S24 ; S25 ; S26 ; S27 ;
S28 ; S29 . Cavity magnonics has now become a new platform for the study of
strong interactions between light and matter, in the context of cavity QED
with magnons. As one of the main advantages, the magnonic system shows an
excellent ability to coherently interact with diverse quantum systems,
including microwave S1 ; S2 ; S3  or optical photons S7 ; S8 ; S9 , phonons
S9b ; S17 ; S26 ; S29 , and superconducting qubits S5 ; S10 ; S23 ; S27 .
Hybrid cavity magnonic systems promise potential applications in quantum-
information processing NakaRev , quantum sensing Yan ; Ruoso ; Ali , and in
searching dark-matter axions DM , to name a few.

In this paper, we show the potential to build a quantum network Kimble ;
Wehner  based on magnonic systems in view of their aforementioned excellent
properties. A future quantum network could be constructed based on single
atoms in optical cavities Rempe , or atomic ensembles following the Duan-
Lukin-Cirac-Zoller protocol DLCZ , or trapped atomic ions Duan , etc. Compared
to these platforms as quantum nodes, where the atomic energy levels are fixed,
a major advantage of magnonic systems lies in the fact that their resonance
frequencies can be continuously adjusted by altering the external magnetic
field. This offers a large flexibility to couple to different quantum systems,
like superconducting qubits, photons, and phonons NakaRev . Therefore, a
quantum network based on magnonic systems shows its unique advantages.
However, a major obstacle for such a magnon-based quantum network is that its
coherence time is limited by its intrinsic loss (typically with damping rate
$\gamma_{m}/2\pi\,\,{\sim}\,1$ MHz), and is in the order of 1 $\mu$s for YIG.
The coherence time can indeed be significantly extended by transferring the
magnonic quantum state to the mechanical mode (i.e., the vibrational phonon
mode) of the same YIG ferrimagnet Jie19 ; JieQST ; Jing ; Sarma , which can
act as a long-lived quantum memory Simon ; Simon2 . However, this local
operation via magnomechanics does not allow to build a quantum network with
its nodes distributed in a long distance.

Here, we show that this obstacle can be eliminated by using light, an optimal
candidate for transmitting quantum information over a long distance, through
which the magnonic system is connected to a distant mechanical system. We, for
the first time, prove that light can connect two separate fields of
optomagnonics and optomechanics, and be used to accomplish some basic
functions of a quantum network, such as quantum state transfer and
entanglement distribution among different nodes of the network Kimble ; Rempe
; cirac . The quantum network with magnonic and mechanical nodes combines the
advantages of both systems, i.e., the great magnonic compatibility and
tunability as well as the long mechanical coherence time. Remarkably, a recent
optomechanical experiment has demonstrated a mechanical coherence time longer
than 100 ms Albert . Specifically, we show that an arbitrary magnonic state,
either quantum or classical, can be transferred to a distant mechanical
resonator by using optical pulses to successively activate the optomagnonic
and optomechanical anti-Stokes processes. This allows the transfer of the
magnonic state to the anti-Stokes optical pulse, and then the mapping of the
pulse state to the mechanical mode. The magnonic state can be stored in the
mechanical mode within its coherence time and retrieved by sending a weak red-
detuned read pulse to the optomechanical cavity, of which the output field
carries the magnonic state. We study the fidelity in this magnon-to-phonon
state transfer process, and show its dependence on the system parameters,
e.g., the pulse strengths and durations, and the transmission loss.

We further show that the magnonic and mechanical nodes can be prepared in a
macroscopic entangled state by using pulses to successively activate the
optomagnonic Stokes and optomechanical anti-Stokes processes. The former
process realizes a two-mode squeezed vacuum state of the magnons and the
pulse, and the latter process maps the pulse state onto the mechanical mode,
thus establishing a nonlocal entangled state between the magnonic and
mechanical nodes.

The paper is organized as follows. In Sec. II, we introduce some basic
interactions in cavity optomagnonics and optomechanics, which are necessities
for realizing our protocol. In Sec. III, we show how to operate the system via
optical pulses such that an arbitrary magnonic state can be transferred to a
distant long-lived mechanical resonator. We further analyse the fidelity in
this state transfer process and show its dependence on the system parameters,
especially on those related to optical pulses. We then study the effects of
the transmission loss of the pulse on the transferred mechanical state and the
fidelity. In Sec. IV, we show how to prepare a nonlocal macroscopic entangled
state between the magnonic and mechanical nodes, and provide a strategy to
detect it. Finally, we draw the conclusions in Sec. V.

##  II Basic interactions in optomagnonics and optomechanics

We start with the introduction of the basic interactions in cavity
optomagnonics and optomechanics that are key building blocks for realizing our
protocol. These include the optomagnonic (optomechanical) two-mode squeezing
and beamsplitter interactions that are used for realizing the entangling and
state-swap operations, respectively. We explicitly show how to operate the two
subsystems to realize these interactions and provide their effective
Hamiltonians. With the successful implementation of these local operations, we
prove in the next section that a remote quantum network based on magnonic and
mechanical systems can be built using optical pulses.

###  II.1 Magnon-induced Brillouin light scattering in optomagnonics

We consider a cavity optomagnonic system of a YIG sphere S7 ; S8 ; S9  that
simultaneously supports a magnetostatic mode of magnons and whispering gallery
modes (WGMs) of optical photons, as depicted in Fig. 1(a). The photons in a
WGM are scattered by the lower-frequency magnons, typically in GHz S7 ; S8 ;
S9 , yielding sideband photons with their frequency shifted by the magnon
frequency. This process is known as the magnon-induced Brillouin light
scattering (BLS). When the scattered photons go into another WGM (the so-
called triple-resonance condition), the BLS scattering probability is then
maximized. This triple resonance can be conveniently achieved by tuning the
magnon frequency realized by altering the strength of the bias magnetic field.
Owing to the selection rule GEB17 ; Papa ; UsamiNJP ; Haigh18  imposed by the
conservation of the angular momenta of WGM photons and magnons, the BLS shows
a pronounced asymmetry in the Stokes and anti-Stokes scattering strengths.
This asymmetry is the basis for realizing the proposals for preparing
macroscopic quantum states of magnons in optomagnonics GEB18 ; SVK19 ; Zhou ;
Xie ; Jie21 ; Heqy .

Figure 1: (a) An optomagnonic system of a YIG sphere supporting two WGMs and a
magnon mode. (b) Mode frequencies of the optomagnonic Stokes BLS. (c) Mode
frequencies of the optomagnonic anti-Stokes BLS.

The magnon-induced BLS is intrinsically a three-wave process, which can be
described by the Hamiltonian

$H=H_{0}+H_{\rm int}+H_{d},$ (1)

where $H_{0}$ is the free Hamiltonian of two WGMs and a magnon mode

$H_{0}/\hbar=\omega_{1}a_{1}^{{\dagger}}a_{1}+\omega_{2}a_{2}^{{\dagger}}a_{2}+\omega_{m}m^{{\dagger}}m,$
(2)

with $a_{j}$ and $m$ ($a_{j}^{{\dagger}}$ and $m^{{\dagger}}$, $j=1,2$) being
the annihilation (creation) operators of the WGMs and magnon mode,
respectively, and $\omega_{i}$ ($i=1,2,m$) being their resonance frequencies,
which satisfy the relation $\omega_{m}\ll\omega_{j}$ and
$|\omega_{1}-\omega_{2}|=\omega_{m}$, imposed by the conservation of energy in
the BLS. The interaction Hamiltonian $H_{\rm int}$ of the three modes is given
by

$H_{\rm
int}/\hbar=G_{0}\big{(}a_{1}^{{\dagger}}a_{2}m^{{\dagger}}+a_{1}a_{2}^{{\dagger}}m\big{)},$
(3)

where $G_{0}$ is the single-photon coupling rate. This coupling is weak due to
the large frequency difference between the optical and magnon modes, but it
can be significantly enhanced by intensely driving one of the WGMs. The
driving Hamiltonian is

$H_{d}/\hbar=iE_{j}\big{(}a_{j}^{{\dagger}}e^{-i\omega_{d}t}-a_{j}e^{i\omega_{d}t}\big{)},$
(4)

where $E_{j}=\\!\sqrt{P_{j}\kappa_{j}^{e}/\hbar\omega_{d}}$ is the coupling
strength between the $j$th WGM (with external decay rate $\kappa_{j}^{e}$) and
the driving field (with frequency $\omega_{d}$ and power $P_{j}$). To maximize
the BLS scattering probability, we resonantly pump either the WGM $a_{1}$ or
$a_{2}$ S7 ; S8 ; S9  (i.e., $\omega_{d}=\omega_{1}$ or $\omega_{2}$) to
selectively activate the anti-Stokes or Stokes scattering, which is
responsible for the optomagnonic state-swap or two-mode squeezing interaction.
Note that the selection rule also causes different optical polarizations of
the two WGMs. Without loss of generality, we assume $a_{2}$ ($a_{1}$) mode to
be the TM (TE) mode of a certain WGM orbit, and $\omega_{2({\rm
TM})}>\omega_{1({\rm TE})}$ due to the geometrical birefringence of the WGM
resonator S7 ; Jie21 . It is worth noting that this is true for WGMs with the
same angular momentum, but when the optical angular momentum changes the
dominant optomagnonic coupling occurs between WGMs satisfying $\omega_{{\rm
TM}}<\omega_{{\rm TE}}$ GEB17 ; UsamiNJP ; Haigh18 .

The Hamiltonian $H$ takes a compact form in the frame rotating at the drive
frequency $\omega_{d}$, which is

$\begin{split}H/\hbar&=\Delta_{1}a_{1}^{{\dagger}}a_{1}+\Delta_{2}a_{2}^{{\dagger}}a_{2}+\omega_{m}m^{{\dagger}}m\\\
&+G_{0}\big{(}a_{1}^{{\dagger}}a_{2}m^{{\dagger}}+a_{1}a_{2}^{{\dagger}}m\big{)}+iE_{j}\big{(}a_{j}^{{\dagger}}-a_{j}\big{)},\end{split}$
(5)

where $\Delta_{j}=\omega_{j}-\omega_{d}$ ($j=1,2$) is the cavity-drive
detuning. We now consider the case where mode $a_{2}$ is resonantly pumped by
a strong optical field, i.e., $\Delta_{2}=0$, and thus
$\Delta_{1}=-\omega_{m}$, cf. Fig. 1(b). This can be realized by, e.g.,
coupling the laser field with a certain polarization to the TM mode of the
anticlockwise WGM orbit S7 . In this case, the strongly driven WGM $a_{2}$ can
be treated classically as a number $\alpha_{2}\equiv\langle
a_{2}\rangle=2E_{2}/\kappa_{2}$, with $\kappa_{2}$ the linewidth (FWHM) of the
mode, and $N_{2}=|\alpha_{2}|^{2}$ is the intra-cavity photon number. The
linearized Hamiltonian in the interaction picture can then be obtained

$H^{\rm St.}_{\rm
int}/\hbar=G_{1}\Big{(}a_{1}^{{\dagger}}m^{{\dagger}}e^{-i(\Delta_{1}+\omega_{m})t}+a_{1}me^{i(\Delta_{1}+\omega_{m})t}\Big{)},$
(6)

where $G_{1}=G_{0}\alpha_{2}$ is the effective coupling rate. Since we
consider a resonant pump, $\Delta_{2}=0$ and thus $\Delta_{1}=-\omega_{m}$,
the above Hamiltonian reduces to

$H^{\rm St.}_{\rm int}=\hbar
G_{1}\big{(}a_{1}^{{\dagger}}m^{{\dagger}}+a_{1}m\big{)},$ (7)

which accounts for the two-mode squeezing interaction between the optical mode
$a_{1}$ and magnon mode $m$, and can be used to prepare optomagnonic entangled
states. This corresponds to the Stokes scattering process, where pump (TM
polarized) photons convert into lower-frequency sideband (TE polarized)
photons by creating magnon excitations. The corresponding quantum Langevin
equations (QLEs), when taking into account the dissipation and input noise of
each mode, are given by

$\begin{split}\dot{a}_{1}&=-\frac{\kappa_{1}}{2}a_{1}-iG_{1}m^{{\dagger}}+\\!\sqrt{\kappa_{1}}a_{1}^{\rm
in},\\\
\dot{m}&=-\frac{\kappa_{m}}{2}m-iG_{1}a_{1}^{{\dagger}}+\\!\sqrt{\kappa_{m}}m^{\rm
in},\end{split}$ (8)

with $\kappa_{1}$ ($\kappa_{m}$) being the linewidth and $a_{1}^{\rm in}$
($m^{\rm in}$) the input noise of the WGM $a_{1}$ (magnon mode). Note that,
for simplicity, we assume the intrinsic decay rate of each WGM
$\kappa_{j}^{i}\ll\kappa_{j}^{e}\simeq\kappa_{j}$ ($j=1,2$), such that for
each WGM we can approximately write a single input noise operator associated
with the total decay rate $\kappa_{j}$.

Similarly, when mode $a_{1}$ is resonantly pumped by a strong field (i.e.,
$\Delta_{1}=0$ and $\Delta_{2}=\omega_{m}$, cf. Fig. 1(c)), e.g., by coupling
the laser field to the TE mode of the anticlockwise WGM orbit S7 , we obtain
the linearized Hamiltonian in the interaction picture

$H^{\rm A.S.}_{\rm int}=\hbar
G_{2}\big{(}a_{2}m^{{\dagger}}+a_{2}^{{\dagger}}m\big{)},$ (9)

where $G_{2}=G_{0}\alpha_{1}$ and $\alpha_{1}=2E_{1}/\kappa_{1}$. This is the
state-swap interaction between the optical mode $a_{2}$ and magnon mode $m$,
and can be used to read out the magnon state by measuring the created anti-
Stokes field $a_{2}$. This anti-Stokes scattering corresponds to the process
where pump (TE polarized) photons convert into higher-frequency anti-Stokes
(TM polarized) photons by annihilating magnons. The Hamiltonian (9) leads to
the following QLEs:

$\begin{split}\dot{a}_{2}&=-\frac{\kappa_{2}}{2}a_{2}-iG_{2}m+\\!\sqrt{\kappa_{2}}a_{2}^{\rm
in},\\\ \dot{m}&=-\frac{\kappa_{m}}{2}m-iG_{2}a_{2}+\\!\sqrt{\kappa_{m}}m^{\rm
in},\end{split}$ (10)

where $a_{2}^{\rm in}$ is the input noise entering the WGM $a_{2}$.

###  II.2 Mechanical motion-induced light scattering in optomechanics

The cavity optomechanical system consists of an optical cavity mode and a
mechanical resonator that are coupled by the radiation pressure OMrmp , as
depicted in Fig. 2(a). The cavity is driven by a laser field that is tuned on
either the red or the blue mechanical sideband for realizing the
optomechanical state-swap or two-mode squeezing interaction. In particular,
the former state-swap interaction will be utilized in the present protocol for
building a magnon-phonon quantum network. We now show how to realize these
operations and derive their effective interaction Hamiltonians. The
Hamiltonian of a typical optomechanical system reads

$\begin{split}H/\hbar=\omega_{c}c^{{\dagger}}c+\omega_{M}b^{{\dagger}}b-gc^{{\dagger}}c(b\,{+}\,b^{{\dagger}})+i\epsilon(c^{{\dagger}}e^{-i\omega_{L}t}{-}ce^{i\omega_{L}t}),\end{split}$
(11)

where $c$ and $b$ ($c^{{\dagger}}$ and $b^{{\dagger}}$) are the annihilation
(creation) operators of the cavity and mechanical modes, respectively,
$\omega_{c}$ and $\omega_{M}$ are their resonance frequencies, $g$ is the
single-photon optomechanical coupling rate, and
$\epsilon=\\!\sqrt{P_{L}\kappa_{c}^{e}/\hbar\omega_{L}}$ is the coupling
strength between the cavity (with external decay rate $\kappa_{c}^{e}$) and
the laser field (with frequency $\omega_{L}$ and power $P_{L}$). This
Hamiltonian leads to the following QLEs, by including the dissipations and
input noises of the two modes, in the frame rotating at the laser frequency

$\begin{split}\dot{c}&=-\Big{(}\frac{\kappa_{c}}{2}+i\Delta_{c}\Big{)}c+igc(b+b^{{\dagger}})+\epsilon+\\!\sqrt{\kappa_{c}}c^{\rm
in},\\\
\dot{b}&=-\Big{(}\frac{\gamma}{2}+i\omega_{M}\Big{)}b+igc^{{\dagger}}c+\\!\sqrt{\gamma}b^{\rm
in},\end{split}$ (12)

where $\Delta_{c}=\omega_{c}-\omega_{L}$ is the cavity-laser detuning,
$\kappa_{c}$ ($\gamma$) is the linewidth and $c^{\rm in}$ ($b^{\rm in}$) is
the input noise of the cavity (mechanical) mode. When the cavity is strongly
driven, the cavity field amplitude $|\langle c\rangle|\gg 1$, which allows us
to linearize the system dynamics around the classical averages (by writing the
operators as $c=\langle c\rangle+\delta c$ and $b=\langle b\rangle+\delta b$)
and obtain the linearized QLEs for the quantum fluctuations

$\begin{split}\delta\dot{c}&=-\Big{(}\frac{\kappa_{c}}{2}+i\tilde{\Delta}_{c}\Big{)}\delta
c+iG(\delta b+\delta b^{{\dagger}})+\\!\sqrt{\kappa_{c}}c^{\rm in},\\\
\delta\dot{b}&=-\Big{(}\frac{\gamma}{2}+i\omega_{M}\Big{)}\delta b+iG(\delta
c+\delta c^{{\dagger}})+\\!\sqrt{\gamma}b^{\rm in},\end{split}$ (13)

with $\tilde{\Delta}_{c}=\Delta_{c}-2g{\rm Re}\langle b\rangle$ being the
effective detuning, which includes the frequency shift caused by the
optomechanical interaction, and $G=g\langle c\rangle$ the effective coupling
rate, where $\langle
c\rangle=\frac{\epsilon}{(\kappa_{c}/2)+i\tilde{\Delta}_{c}}$. Note that in
getting QLEs (13), we choose a phase reference such that $\langle c\rangle$
(and thus $G$) is real, and the expression of $\langle c\rangle$ is the
steady-state solution.

Figure 2: (a) A cavity optomechanical system: a cavity mode couples to a
mechanical resonator via radiation pressure. (b) A red-detuned drive field to
activate the optomechanical anti-Stokes process for realizing state swap. (c)
A blue-detuned drive field to activate the optomechanical Stokes process for
realizing two-mode squeezing.

To see more clearly how to operate the system to realize the two basic
optomechanical interactions, we move to another interaction picture by
introducing the slowly moving operators $\delta\tilde{c}(t)=\delta
c(t)e^{i\tilde{\Delta}_{c}t}$ and $\delta\tilde{b}(t)=\delta
b(t)e^{i\omega_{M}t}$. The QLEs then take the following form in the
interaction picture (for simplicity we remove the tilde signs on the
operators):

$\begin{split}\delta\dot{c}&={-}\frac{\kappa_{c}}{2}\delta c+iG\Big{[}\delta
b\,e^{i(\tilde{\Delta}_{c}-\omega_{M})t}{+}\delta
b^{{\dagger}}e^{i(\tilde{\Delta}_{c}+\omega_{M})t}\Big{]}+\\!\sqrt{\kappa_{c}}c^{\rm
in},\\\ \delta\dot{b}&={-}\frac{\gamma}{2}\delta b+iG\Big{[}\delta
c\,e^{i(\omega_{M}-\tilde{\Delta}_{c})t}{+}\delta
c^{{\dagger}}e^{i(\omega_{M}+\tilde{\Delta}_{c})t}\Big{]}+\\!\sqrt{\gamma}b^{\rm
in}.\end{split}$ (14)

For a red-detuned driving field, $\tilde{\Delta}_{c}=\omega_{M}$ (cf. Fig.
2(b)), we obtain the approximate equations

$\begin{split}\delta\dot{c}&\approx-\frac{\kappa_{c}}{2}\delta c+iG\delta
b+\\!\sqrt{\kappa_{c}}c^{\rm in},\\\
\delta\dot{b}&\approx-\frac{\gamma}{2}\delta b+iG\delta
c+\\!\sqrt{\gamma}b^{\rm in},\end{split}$ (15)

which corresponds to the linearized interaction Hamiltonian

$H^{\rm A.S.}_{\rm int}=-\hbar G(c^{{\dagger}}b+cb^{{\dagger}}),$ (16)

and the process of anti-Stokes scattering for realizing the state-swap
interaction between the mechanics and the cavity field. While for a blue-
detuned driving field, $\tilde{\Delta}_{c}=-\omega_{M}$ (cf. Fig. 2(c)), we
get

$\begin{split}\delta\dot{c}&\approx-\frac{\kappa_{c}}{2}\delta c+iG\delta
b^{{\dagger}}+\\!\sqrt{\kappa_{c}}c^{\rm in},\\\
\delta\dot{b}&\approx-\frac{\gamma}{2}\delta b+iG\delta
c^{{\dagger}}+\\!\sqrt{\gamma}b^{\rm in},\end{split}$ (17)

which corresponds to the interaction Hamiltonian

$H^{\rm St.}_{\rm int}=-\hbar G(c^{{\dagger}}b^{{\dagger}}+cb),$ (18)

and the process of Stokes scattering for achieving the two-mode squeezing
interaction between the mechanical and cavity modes. Note that in deriving
Eqs. (15) and (17) we assume $\gamma,\,\kappa_{c},\,G\ll\omega_{M}$, such that
the non-resonant fast oscillating terms play a negligible role and can be
neglected.

Figure 3: Sketch of the distant magnon-to-phonon state transfer protocol (a)
and the corresponding mode frequencies (c). The nonlocal magnon-phonon
entanglement protocol (b) and the associated mode frequencies (d).

##  III Distant Magnon-to-phonon state transfer

Equipped with all the necessary tools, we now proceed to describe our
protocol. Note that the cavity optomagnonic system using a YIG sphere
currently works in the weak coupling regime, where the enhanced optomagnonic
coupling rate (potentially in MHz) is much smaller than the decay rate of the
WGM (from $10^{2}$ MHz to GHz), $G_{j}\ll\kappa_{j}$ S7 ; S8 ; S9 . The large
cavity decay disables many quantum protocols that require the cooperativity
${\cal C}=G_{j}^{2}/(\kappa_{j}\kappa_{m})>1$, considering also
$\kappa_{m}/2\pi$ is hardly below 1 MHz for YIG S1 ; S2 ; S3 . However, this
deficiency of the system can be evaded by using fast optical pulses Simon16 ;
Simon18 ; Jie21 ; Xie ; Heqy .

In the first part, we aim to realize a distant magnon-to-phonon (quantum)
state transfer. Specifically, as shown in Fig. 3(a), by using laser pulses to
successively activate the anti-Stokes scatterings in the optomagnonic and
optomechanical systems, an arbitrary magnonic state can be transferred to a
mechanical resonator that can have a much longer coherence time. We have
introduced the interaction Hamiltonian (10) of the optomagnonic anti-Stokes
scattering in Sec. II A, associated with the process where a TE polarized
pulse couples to a WGM and generates TM polarized anti-Stokes photons in
another WGM by annihilating magnons. We consider laser pulses with duration
much shorter than the magnon lifetime note . In this case, the dissipation of
the magnon mode within the pulse duration is negligibly small, and we thus
neglect it for simplicity. This leads to the following QLEs during the pulse
interaction:

$\begin{split}\dot{a}_{2}&=-\frac{\kappa_{2}}{2}a_{2}-iG_{2}m+\\!\sqrt{\kappa_{2}}a_{2}^{\rm
in},\\\ \dot{m}&=-iG_{2}a_{2}.\end{split}$ (19)

To simplify the model, we consider a flattop pulse and thus a constant
coupling $G_{2}$ during the pulse. Given also the fact that a weak coupling
$G_{2}\ll\kappa_{2}$, one can then adiabatically eliminate the cavity, and
obtain
$a_{2}\simeq\frac{2}{\kappa_{2}}\big{(}-iG_{2}m+\\!\sqrt{\kappa_{2}}a_{2}^{\rm
in}\big{)}$. By using the input-output relation $a_{2}^{\rm
out}\,{=}\,\sqrt{\kappa_{2}}a_{2}-a_{2}^{\rm in}$ collett , we obtain

$\begin{split}a_{2}^{\rm out}&=-i\sqrt{2{\cal G}_{2}}m+a_{2}^{\rm in},\\\
\dot{m}&=-{\cal G}_{2}m-i\sqrt{2{\cal G}_{2}}a_{2}^{\rm in},\end{split}$ (20)

where ${\cal G}_{2}\equiv 2G_{2}^{2}/\kappa_{2}$. Following Hofer , we define
a set of normalized temporal modes for the WGM driven by a pulse of duration
$\tau_{2}$

$\begin{split}A_{2}^{\rm in}(\tau_{2})&=\sqrt{\frac{2{\cal G}_{2}}{e^{2{\cal
G}_{2}\tau_{2}}-1}}\int_{0}^{\tau_{2}}e^{{\cal G}_{2}s}a_{2}^{\rm
in}(s)\,ds,\\\ A_{2}^{\rm out}(\tau_{2})&=\sqrt{\frac{2{\cal
G}_{2}}{1-e^{-2{\cal G}_{2}\tau_{2}}}}\int_{0}^{\tau_{2}}e^{-{\cal
G}_{2}s}a_{2}^{\rm out}(s)\,ds,\end{split}$ (21)

which satisfy the canonical commutation relation
$[A^{j},\,\,A^{j{\dagger}}]\,{=}\,1,j\,{=}\,\\{{\rm in},\,\,{\rm out}\\}$.
Therefore, by integrating (20) we obtain the following solutions (see
Appendix):

$\begin{split}A_{2}^{\rm out}(\tau_{2})&=-i\sqrt{1-e^{-2{\cal
G}_{2}\tau_{2}}}m(0)+e^{-{\cal G}_{2}\tau_{2}}A_{2}^{\rm in}(\tau_{2}),\\\
m(\tau_{2})&=e^{-{\cal G}_{2}\tau_{2}}m(0)-i\sqrt{1-e^{-2{\cal
G}_{2}\tau_{2}}}A_{2}^{\rm in}(\tau_{2}).\end{split}$ (22)

From these solutions, we can extract a propagator $L_{2}(\tau_{2})$ that
satisfies $A_{2}^{\rm out}(\tau_{2})=L_{2}^{{\dagger}}(\tau_{2})\,A_{2}^{\rm
in}(\tau_{2})\,L_{2}(\tau_{2})$ and
$m(\tau_{2})=L_{2}^{{\dagger}}(\tau_{2})\,m(0)\,L_{2}(\tau_{2})$, given by Xie

$L_{2}(\tau_{2})=e^{-i\sqrt{S^{\prime}}A_{2}^{\rm in{\dagger}}m}e^{{\cal
G}_{2}\tau_{2}(A_{2}^{\rm in{\dagger}}A_{2}^{\rm
in}-m^{{\dagger}}m)}e^{i\sqrt{S^{\prime}}A_{2}^{\rm in}m^{{\dagger}}},$ (23)

where $S^{\prime}=Se^{2{\cal G}_{2}\tau_{2}}$, with $S=1-e^{-2{\cal
G}_{2}\tau_{2}}$ ($0<S<1$) being the optomagnonic state conversion efficiency,
which depends on the pulse strength and duration. It is clear that when ${\cal
G}_{2}\tau_{2}\gg 1$, $S\to 1$, and we thus get $A_{2}^{\rm
out}(\tau_{2})\simeq-im(0)$, which implies that the magnon state is perfectly
transferred to the optical mode, apart from a phase difference $-i$. A similar
mechanism by using the light anti-Stokes scattering has been adopted to read
out the mechanical state in optomechanics Hofer ; tjk ; jie18 ; Simon16 ;
Simon18 .

We assume that the optomagnonic system is in an initial state

$\rho_{m,2}(0)=\sum_{n,s=0}^{\infty}c_{n,s}|n\rangle\langle
s|_{m}\otimes|0\rangle\langle 0|_{2},$ (24)

before sending the TE polarized pulse to couple to the WGM. To be generic, the
magnon mode is assumed in an arbitrary state, which can be expanded in the
Fock-state basis with arbitrary real coefficients $c_{n,s}\geq 0$, i.e.,
$\rho_{m}(0)=\sum_{n,s=0}^{\infty}c_{n,s}|n\rangle\langle s|_{m}$ jie10 , and
the WGM $a_{2}$ in vacuum state. As discussed, the TE polarized pulse (with
duration $\tau_{2}$ and strength yielding an effective coupling $G_{2}$)
activates the anti-Stokes BLS, associated with the time-evolution propagator
$L_{2}(\tau_{2})$. The system, at the end of the pulse, evolves to be

$\rho_{m,2}(\tau_{2})=\sum_{n,s=0}^{\infty}c_{n,s}(-i)^{n}i^{s}S^{\frac{n+s}{2}}|0\rangle\langle
0|_{m}\otimes|n\rangle\langle s|_{2}.$ (25)

Clearly, the initial magnon state $\rho_{m}(0)$ is transferred to the TM
polarized pulse (anti-Stokes photons) in the state $\rho_{\rm
TM}(\tau_{2})=\sum_{n,s=0}^{\infty}c_{n,s}(-i)^{n}i^{s}S^{\frac{n+s}{2}}|n\rangle\langle
s|_{2}$. The fidelity in this state transfer is reduced owing to a nonunity
conversion efficiency $S<1$ (apart from a phase difference $(-i)^{n}i^{s}$).

The TM polarized pulse then transmits, through a fiber, to a distant
optomechanical system and resonantly drives the optomechanical cavity, see
Figs. 3(a), (c). Meantime, the cavity is driven by another red-detuned pulse,
which activates the optomechanical anti-Stokes process for realizing the
state-swap between the cavity field and the mechanics. Such a configuration
has been employed to transfer the state of an electromagnetic field to a
mechanical resonator Jie19 ; JieQST ; jie18 ; zoller ; tian . Owing to the
similarity between the optomagnonic and optomechanical anti-Stokes scatterings
(in their interaction Hamiltonians (9) and (16)), following the same approach
as from Eq. (19) to Eq. (23), we can extract a propagator associated with the
pulse (with duration $\tau_{b}$) that activates the optomechanical anti-Stokes
process

$L_{\rm oM}(\tau_{b})=e^{i\sqrt{W^{\prime}}C^{\rm in}b^{{\dagger}}}e^{{\cal
G}\tau_{b}(b^{{\dagger}}b-C^{\rm in{\dagger}}C^{\rm
in})}e^{-i\sqrt{W^{\prime}}C^{\rm in{\dagger}}b},$ (26)

where $C^{\rm in}$ denotes the temporal mode of the input field (entering the
optomechanical cavity), ${\cal G}\equiv 2G^{2}/\kappa_{c}$, and
$W^{\prime}=We^{2{\cal G}\tau_{b}}$, with $W=1-e^{-2{\cal G}\tau_{b}}$ being
the optomechanical state conversion efficiency, $0<W<1$. Similarly, we assume
the pulse duration to be much shorter than the mechanical lifetime and a weak
coupling $G\ll\kappa_{c}$, which can be easily satisfied because of a
relatively longer mechanical lifetime OMrmp .

We further assume that the mechanical mode is prepared in the ground state
$|0\rangle_{M}$ (for a GHz resonator requiring a bath temperature of tens of
mK Simon16 ; Simon18 ). Alternatively, the mechanical resonator can also be
precooled to its ground state using a red-detuned light chen . The previously
generated TM polarized pulse now acts as the input field into the
optomechanical cavity, and the evolution of the system can be solved by
applying the propagator $L_{\rm oM}(\tau_{b})$ onto the initial state notee

$\rho_{\rm
oM}(0)=\sum_{n,s=0}^{\infty}c_{n,s}(-i)^{n}i^{s}S^{\frac{n+s}{2}}|n\rangle\langle
s|_{o}\otimes|0\rangle\langle 0|_{M}.$ (27)

At the end of the pulse, we obtain the state $\rho_{\rm oM}(\tau_{b})=L_{\rm
oM}(\tau_{b})\rho_{\rm oM}(0)L_{\rm oM}^{{\dagger}}(\tau_{b})$, given by

$\rho_{\rm
oM}(\tau_{b})=\sum_{n,s=0}^{\infty}c_{n,s}(SW)^{\frac{n+s}{2}}|0\rangle\langle
0|_{o}\otimes|n\rangle\langle s|_{M}.$ (28)

It shows that the state of the TM polarized pulse $\rho_{\rm TM}(\tau_{2})$ is
transferred to the mechanical resonator, which is in the state
$\rho_{M}(\tau_{b})=\sum_{n,s=0}^{\infty}c_{n,s}(SW)^{\frac{n+s}{2}}|n\rangle\langle
s|_{M}$, with a reduced fidelity due to a nonunity conversion efficiency
$W<1$.

Looking at the whole process, after two state-swap operations (magnon-to-
photon and photon-to-phonon), an arbitrary magnon state $\rho_{m}(0)$ is
successfully transferred to the mechanical mode in the state
$\rho_{M}(\tau_{b})$. The fidelity is determined by the product of the two
conversion efficiencies $S$ and $W$ in the two anti-Stokes processes. We
provide two concrete examples: When the magnon mode is initially in a Fock
state $|\phi\rangle_{m}=|n\rangle$, we obtain the final transferred mechanical
state $|\phi^{\prime}\rangle_{M}=(SW)^{\frac{n}{2}}|n\rangle$. The fidelity is
${\cal F}=|\langle\phi|\phi^{\prime}\rangle|^{2}=(SW)^{n}$, which reduces with
an increasing $n$. For an initial superposition state
$|\phi\rangle_{m}=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$, we get
$|\phi^{\prime}\rangle_{M}=\frac{1}{\sqrt{2}}\big{(}|0\rangle+\sqrt{SW}|1\rangle\big{)}$,
and the fidelity ${\cal F}=\frac{1}{4}\big{(}1+\sqrt{SW}\big{)}^{2}$. For
perfect state conversions $S,W\to 1$, so the fidelity ${\cal F}\to 1$. Note
that, because of the nonunity state conversion efficiencies in actual
experiments, the experiment will be repeated many times using a sequence of
pulses. The fidelity then can be interpreted as the probability of the
successful/perfect magnon-to-phonon state transfer in a single experiment run
Simon2 . The mechanical state can be stored within the mechanical coherence
time and retrieved by sending a weak red-detuned read pulse to the cavity and
measuring the cavity output field Simon ; Simon2 ; Simon16 ; Simon18 .

It should be noted that, for simplifying the calculations, we adopt laser
pulses with duration much shorter than the magnon lifetime $\sim$ 1 $\mu$s for
the optomagnonic system. This allows us to neglect the loss of the magnons.
However, the small pulse duration also reduces the conversion efficiency and
thus the fidelity in the state transfer. This can be compensated by increasing
the pulse strength or the single-photon coupling rate $G_{0}$, e.g., by
reducing the mode volumes and increasing the mode overlap of the magnon and
optical fields, purifying and doping YIG Greek , coupling WGMs to a magnetic
vortex SVK18 , and utilizing the epsilon-near-zero medium SVK21 , etc. Let us
estimate the optomagnonic conversion efficiency using promising parameters.
Taking a cavity decay rate $\kappa_{2}/2\pi\,{\sim}\,500$ MHz, an effective
coupling $G_{2}/2\pi\,{\sim}\,10$ MHz S8 ; Flat , and a pulse duration
$\tau_{2}=40$ ns, we obtain ${\cal
G}_{2}\tau_{2}=\frac{2G_{2}^{2}}{\kappa_{2}}\tau_{2}\simeq 0.10$, and a
moderate efficiency $S=1-e^{-2{\cal G}_{2}\tau_{2}}\simeq 0.18$.

For the mechanical system, because of its much longer lifetime, longer pulses
could be used to increase the conversion efficiency. We adopt the parameters
from an optomechanical experiment Simon16 : a mechanical mode of frequency
$\omega_{M}/2\pi\,\,{=}\,5.3$ GHz and damping rate $\gamma/2\pi\,\,{=}\,4.8$
kHz (corresponding to lifetime $2\pi/\gamma\,\,{\simeq}\,\,0.2$ ms), a cavity
decay rate $\kappa_{c}/2\pi\,\,{=}\,1.3$ GHz, a pulse duration
$\tau_{b}\,{=}\,55$ ns, and an effective coupling $G/2\pi\,\,{=}\,50$ MHz.
These parameters yield ${\cal
G}\tau_{b}\,{=}\,\frac{2G^{2}}{\kappa_{c}}\tau_{b}\,{\simeq}\,1.33$, and a
high conversion efficiency $W\,{=}\,1-e^{-2{\cal G}\tau_{b}}\simeq 0.93$. Note
that there are many different types of optomechanical devices (see Fig.7 and
Table II in OMrmp  for their characteristic parameters), among which a high
conversion efficiency together with a long mechanical lifetime is preferred
for our protocol.

Figure 4: The transmission loss modeled by a beamsplitter (BS), of which the
reflection denotes the loss and the transmission represents the pulse after
suffering the loss.

###  III.1 Effect of the linear loss in pulse transmission

In the preceding section, for simplicity, we neglect the transmission loss of
the pulse from the magnonic to the mechanical system. However, for building a
long-distance quantum network, this loss can be significant. The transmission
loss is linear with the transmission distance, and thus can be modeled by a
linear beamsplitter ULeo , as depicted in Fig. 4. The pulse enters one input
port of the beamsplitter with the other input in vacuum, of which the
reflected part denotes the loss and the transmitted part represents the pulse
after suffering the loss.

Including the loss in the transmission time $\Delta t$, the generated TM pulse
state $\rho_{\rm TM}(\tau_{2})$ (for an arbitrary magnon state) turns into the
following state when reaching the optomechanical cavity jie10 :

$\begin{split}\rho_{\rm TM}&(\tau_{2}+\Delta
t)=\sum_{n,s=0}^{\infty}c_{n,s}(-i)^{n}i^{s}S^{\frac{n+s}{2}}\times\\\
&\,\,\sum_{m=0}^{{\rm
min}(n,s)}\\!\\!\sqrt{\frac{n!s!}{(m!)^{2}(n-m)!(s-m)!}}R^{m}T^{\frac{n+s}{2}-m}|n{-}m\rangle\langle
s{-}m|_{2},\end{split}$ (29)

where $R$ ($T$) is the reflectance (transmittance) of the beamsplitter, and we
assume a lossless beamsplitter $R+T=1$. Then the initial state of the
optomechanical system, before activating the optomechanical anti-Stokes
process, is notee

$\rho^{\prime}_{\rm oM}(0)=\rho_{\rm TM}(\tau_{2}+\Delta
t)\otimes|0\rangle\langle 0|_{M}.$ (30)

We obtain the state $\rho^{\prime}_{\rm oM}(\tau_{b})=L_{\rm
oM}(\tau_{b})\rho^{\prime}_{\rm oM}(0)L_{\rm oM}^{{\dagger}}(\tau_{b})$ soon
after the state-swap interaction, which is

$\begin{split}\rho^{\prime}_{\rm
oM}(\tau_{b})&=\sum_{n,s=0}^{\infty}c_{n,s}S^{\frac{n+s}{2}}\sum_{m=0}^{{\rm
min}(n,s)}\\!\\!\sqrt{\frac{n!s!}{(m!)^{2}(n-m)!(s-m)!}}\\\ &\times
R^{m}(TW)^{\frac{n+s}{2}-m}|0\rangle\langle 0|_{o}\otimes|n-m\rangle\langle
s-m|_{M}.\end{split}$ (31)

The above state does not look intuitive. Let us consider some specific cases
to see the physics more clearly. For an initial magnon Fock state
$|\phi\rangle_{m}=|n\rangle$, Eq. (31) yields the following transferred
mechanical state after the transmission loss:

$\begin{split}\rho^{\prime}_{M}(\tau_{b})&=S^{n}\sum_{m=0}^{n}\frac{n!}{m!(n-m)!}R^{m}(TW)^{n-m}|n{-}m\rangle\langle
n{-}m|_{M}.\end{split}$ (32)

More specifically, for $n=1$, i.e., an initial single-magnon state
$|1\rangle_{m}$, we get

$\begin{split}\rho^{\prime}_{M}(\tau_{b})=S\Big{(}TW|1\rangle\langle
1|_{M}+R|0\rangle\langle 0|_{M}\Big{)}.\end{split}$ (33)

The fidelity between the initial magnon state and the transferred mechanical
state is thus

${\cal F}=\langle\phi|\rho^{\prime}_{M}(\tau_{b})|\phi\rangle=STW,$ (34)

which agrees with our previous result for a magnon Fock state ${\cal
F}\,{=}\,\,(SW)^{n}$ ($n\,\,{=}\,\,1$) for a lossless transmission
$T\,\,{=}\,\,1$. Apparently, the fidelity is proportional to the two
conversion efficiencies and the proportion of the transmitted pulse after the
loss. Considering fiber loss of 0.2 dB/km at telecom wavelengths, for a
distance of 1 km (10 km), the total fiber loss is 0.2 dB (2 dB), corresponding
to $T\simeq 0.955$ (0.631). Taking the two realistic conversion efficiencies
$S\,\,{=}\,\,0.18$ and $W\,\,{=}\,\,0.93$ estimated in the preceding section,
we obtain fidelity ${\cal F}\,\,{=}\,\,STW\,\,{=}\,\,0.16$ (0.10) for
transferring a single-magnon state $|1\rangle_{m}$ to a long-lived mechanical
mode. This means that for 100 runs of the experiment, about 16 (10) times the
mechanical mode is in the single-phonon state $|1\rangle_{M}$.

For the initial superposition state
$|\phi\rangle_{m}=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$, using the result
of Eq. (31), we obtain the mechanical state

$\begin{split}\rho^{\prime}_{M}(\tau_{b})=\frac{1}{2}&\Big{[}\big{(}1+SR\big{)}|0\rangle\langle
0|_{M}+\\!\sqrt{STW}|0\rangle\langle 1|_{M}\\\ &+\\!\sqrt{STW}|1\rangle\langle
0|_{M}+STW|1\rangle\langle 1|_{M}\Big{]}.\end{split}$ (35)

The fidelity is then

${\cal F}=\frac{1}{4}\Big{(}1+SR+2\sqrt{STW}+STW\Big{)},$ (36)

and in the loss-free limit $T\to 1$ ($R\to 0$), it becomes ${\cal
F}=\frac{1}{4}\big{(}1+\sqrt{SW}\big{)}^{2}$, which agrees with the result in
the preceding section for a lossless transmission.

##  IV Nonlocal macroscopic Magnon-phonon entanglement

Under certain circumstances, the quantum network requires its nodes to be
entangled Kimble ; Wehner  and many quantum protocols, like quantum repeaters
Gisin  and teleportation Simon2 , require quantum entanglement. In this
section, we show that our system also allows to prepare a nonlocal macroscopic
magnon-phonon entangled state using optical pulses. Specifically, laser pulses
are sent to successively activate the optomagnonic Stokes scattering and the
optomechanical anti-Stokes scattering, see Figs. 3(b), (d). The former
generates an entangled state of the magnons in YIG and the Stokes photons in
the pulse, and the latter maps the pulse state to the mechanical resonator,
and thus a nonlocal magnon-phonon entangled state is established. The maximum
distance between the two entangled subsystems is determined by the relatively
short magnon lifetime, beyond which the magnon state degrades and the
entanglement dies out.

We introduce the interaction Hamiltonian (8) accounting for the optomagnonic
Stokes scattering. As before, we neglect the magnon dissipation during the
short pulse, and obtain the QLEs

$\begin{split}\dot{a}_{1}&=-\frac{\kappa_{1}}{2}a_{1}-iG_{1}m^{{\dagger}}+\\!\sqrt{\kappa_{1}}a_{1}^{\rm
in},\\\ \dot{m}&=-iG_{1}a_{1}^{{\dagger}}.\end{split}$ (37)

By adiabatically eliminating the cavity field, we get
$a_{1}\simeq\frac{2}{\kappa_{1}}\big{(}-iG_{1}m^{{\dagger}}+\\!\sqrt{\kappa_{1}}a_{1}^{\rm
in}\big{)}$, and using the input-output relation $a_{1}^{\rm
out}=\sqrt{\kappa_{1}}a_{1}-a_{1}^{\rm in}$, we obtain

$\begin{split}a_{1}^{\rm out}&=-i\sqrt{2{\cal G}_{1}}m^{{\dagger}}+a_{1}^{\rm
in},\\\ \dot{m}&={\cal G}_{1}m-i\sqrt{2{\cal G}_{1}}a_{1}^{\rm
in{\dagger}},\end{split}$ (38)

where ${\cal G}_{1}\equiv 2G_{1}^{2}/\kappa_{1}$. Again, we define the
normalized temporal modes for the WGM driven by a pulse of duration $\tau_{1}$

$\begin{split}A_{1}^{\rm in}(\tau_{1})&=\sqrt{\frac{2{\cal
G}_{1}}{1-e^{-2{\cal G}_{1}\tau_{1}}}}\int_{0}^{\tau_{1}}e^{-{\cal
G}_{1}s}a_{1}^{\rm in}(s)\,ds,\\\ A_{1}^{\rm
out}(\tau_{1})&=\sqrt{\frac{2{\cal G}_{1}}{e^{2{\cal
G}_{1}\tau_{1}}-1}}\int_{0}^{\tau_{1}}e^{{\cal G}_{1}s}a_{1}^{\rm
out}(s)\,ds.\end{split}$ (39)

By integrating (38), we obtain

$\begin{split}A_{1}^{\rm out}(\tau_{1})&=-i\sqrt{e^{2{\cal
G}_{1}\tau_{1}}-1}m^{{\dagger}}(0)+e^{{\cal G}_{1}\tau_{1}}A_{1}^{\rm
in}(\tau_{1}),\\\ m(\tau_{1})&=e^{{\cal G}_{1}\tau_{1}}m(0)-i\sqrt{e^{2{\cal
G}_{1}\tau_{1}}-1}A_{1}^{\rm in{\dagger}}(\tau_{1}),\end{split}$ (40)

from which a propagator $L_{1}(\tau_{1})$ is extracted, satisfying $A_{1}^{\rm
out}(\tau_{1})=L_{1}^{{\dagger}}(\tau_{1})\,A_{1}^{\rm
in}(\tau_{1})\,L_{1}(\tau_{1})$ and
$m(\tau_{1})=L_{1}^{{\dagger}}(\tau_{1})\,m(0)\,L_{1}(\tau_{1})$, which is tjk

$L_{1}(\tau_{1})=e^{-i\tanh r\,A_{1}^{\rm in{\dagger}}m^{{\dagger}}}\cosh
r^{(-1-A_{1}^{\rm in{\dagger}}A_{1}^{\rm in}-m^{{\dagger}}m)}e^{i\tanh
r\,A_{1}^{\rm in}m},$ (41)

where we introduce the squeezing parameter $r$ through $\cosh r=e^{{\cal
G}_{1}\tau_{1}}$ and $\tanh r=\sqrt{1-e^{-2{\cal G}_{1}\tau_{1}}}$. It is
clear that the squeezing $r$ increases with the product ${\cal
G}_{1}\tau_{1}$, i.e., the product of the pulse strength and duration.

For an initial state $|0\rangle_{m}|0\rangle_{1}$ nnnot , the optomagnonic
system is prepared, at the end of the pulse, in the state

$|\phi\rangle_{m,1}(\tau_{1})=(\cosh r)^{-1}\sum_{n=0}^{\infty}(\tanh
r)^{n}|n,n\rangle_{m,1},$ (42)

which is a two-mode squeezed vacuum state (TMSVS) of the magnon mode and the
TE polarized pulse (Stokes photons). The entanglement of this state is linked
to the squeezing parameter as the logarithmic negativity $E_{N}=2r$. Using the
parameters $\kappa_{1}/2\pi\,{\sim}\,500$ MHz, $G_{1}/2\pi\,{\sim}\,10$ MHz S8
; Flat , and a pulse duration $\tau_{1}\,{=}\,30$ ns, we obtain ${\cal
G}_{1}\tau_{1}\,{\simeq}\,0.075$, and a squeezing parameter $r\simeq 0.39$,
and the entanglement $E_{N}\simeq 0.78$.

Figure 5: Magnon-phonon entanglement (logarithmic negativity $E_{N}$) versus
the squeezing $r$ for different conversion efficiencies $W$. From upper to
lower curves: $W=1$, 0.8, 0.5, and 0.2.

The TE pulse then transmits to the optomechanical system and resonantly drives
the optical cavity, which is simultaneously driven by another red-detuned
pulse for realizing the optomechanical state-swap operation (cf. Fig. 3(d)).
The corresponding propagator $L_{\rm oM}(\tau_{b})$ has been introduced in Eq.
(26). As in the preceding section, the mechanical resonator is prepared in the
ground state. By applying the propagator $L_{\rm oM}(\tau_{b})$ onto the
initial state

$|\phi\rangle_{m,1,M}(0)=(\cosh r)^{-1}\sum_{n=0}^{\infty}(\tanh
r)^{n}|n,n\rangle_{m,1}\otimes|0\rangle_{M},$ (43)

we obtain the joint state $|\phi\rangle_{m,1,M}(\tau_{b})=L_{\rm
oM}(\tau_{b})|\phi\rangle_{m,1,M}(0)$, at the end of the red-detuned pulse,
given by

$|\phi\rangle_{m,1,M}(\tau_{b})=(\cosh r)^{-1}\sum_{n=0}^{\infty}(i\tanh
r)^{n}W^{\frac{n}{2}}|n,0,n\rangle_{m,1,M}.$ (44)

By tracing over the optical mode and writing in the density matrix form, we
get

$\rho_{m,M}(\tau_{b})=(\cosh r)^{-2}\sum_{n=0}^{\infty}(\tanh
r)^{2n}W^{n}|n,n\rangle\langle n,n|_{m,M},$ (45)

which, in the limit of a unity conversion efficiency $W\,{\to}\,1$ (${\cal
G}\tau_{b}\gg 1$), becomes exactly the TMSVS of the magnon and mechanical
modes, and thus the magnon and mechanical nodes are remotely entangled. For a
realistic efficiency $W\simeq 0.93$ estimated in the preceding section, the
entanglement is only slightly reduced. By comparing the states in Eqs. (42)
and (45), we see that the nonunity conversion efficiency plays a role in
degrading the squeezing and thus the entanglement. This can be seen by making
the replacement $\tanh r^{\prime}=\tanh r\sqrt{W}$, where $r^{\prime}$ is the
effective squeezing including the effect of the nonunity conversion
efficiency, and $r^{\prime}\leq r$. In Fig. 5, we show the entanglement
(logarithmic negativity $E_{N}=2r^{\prime}$) versus the squeezing $r$ for
different conversion efficiencies $W$.

We remark that the survival time of the entanglement is determined by the
relatively short magnon lifetime $\sim 1$ $\mu$s, during which the
transmission loss of the pulse is negligibly small. Therefore, in the
entanglement protocol we do not include the transmission loss. The
entanglement can be verified by activating the optomagnonic and optomechanical
anti-Stokes processes, which transfer the magnonic and mechanical states to
their respective anti-Stokes photons. By measuring the anti-Stokes photons,
one can build a joint magnon-phonon covariance matrix, from which the
entanglement can be verified and quantified. This approach has been used to
measure the optomechanical John1  and mechanical John2  entanglement in
optomechanics.

##  V Conclusion

We present a protocol for building a quantum network based on magnonic and
mechanical systems. The magnonic and mechanical nodes are connected by optical
pulses. We show that the quantum network can realize (both quantum and
classical) state transfer from the magnonic node to the long-lived mechanical
node, taking into account the transmission loss for a long-distance network.
We also show that the two systems can be prepared in a nonlocal macroscopic
entangled state. All these utilize the Stokes and anti-Stokes light
scatterings in optomagnonics and optomechanics, and optical pulses to transfer
quantum states or distribute quantum correlations. As is known, quantum state
transfer and entanglement distribution among different nodes are basic
functions for a quantum network Kimble ; Rempe ; cirac . Our work shows the
possibility to connect the seemingly separate research fields of optomagnonics
and optomechanics, and to build a quantum network based on magnonic systems,
which can be readily extended to a complex network with various nodes by
coupling to other quantum systems, e.g., long-lived phonons, optical/microwave
photons, and superconducting qubits NakaRev . We expect our work may offer a
promising vision for the realization of a hybrid quantum network based on
magnonic systems, and find potential applications in quantum-information
science and in the study of macroscopic quantum states.

## Acknowledgments

We thank S. Gröblacher, D. Vitali, and H. Xie for helpful discussions. This
work is supported by the National Natural Science Foundation of China (Grants
Nos. U1801661, 11934010, 12174329), Zhejiang Province Program for Science and
Technology (Grant No. 2020C01019), and the Fundamental Research Funds for the
Central Universities (No. 2021FZZX001-02).

## APPENDIX

In this section, we provide details on the derivation of the solutions in Eq.
(22), which account for the optomagnonic state-swap operation. By integrating
the second equation in Eq. (20), we obtain the following solution

$m(\tau_{2})=e^{-{\cal G}_{2}\tau_{2}}m(0)-i\sqrt{2{\cal G}_{2}}\,e^{-{\cal
G}_{2}\tau_{2}}\\!\\!\int_{0}^{\tau_{2}}e^{{\cal G}_{2}s}a_{2}^{\rm
in}(s)\,ds.$ (A1)

Using the definition of the temporal mode $A_{2}^{\rm in}(t)$ provided in Eq.
(21), the above equation can be rewritten as

$m(\tau_{2})=e^{-{\cal G}_{2}\tau_{2}}m(0)-i\sqrt{1-e^{-2{\cal
G}_{2}\tau_{2}}}A_{2}^{\rm in}(\tau_{2}),$ (A2)

that is the second equation in Eq. (22).

The second equation of Eq. (20) can also be written in the form of

$a_{2}^{\rm in}=\frac{1}{-i\sqrt{2{\cal G}_{2}}}(\dot{m}+{\cal G}_{2}m).$ (A3)

Substituting it into the first equation of Eq. (20), we obtain

$\dot{m}={\cal G}_{2}m-i\sqrt{2{\cal G}_{2}}a_{2}^{\rm out},$ (A4)

and its solution is given by

$m(\tau_{2})=e^{{\cal G}_{2}\tau_{2}}m(0)-i\sqrt{2{\cal G}_{2}}\,e^{{\cal
G}_{2}\tau_{2}}\\!\\!\int_{0}^{\tau_{2}}e^{-{\cal G}_{2}s}a_{2}^{\rm
out}(s)\,ds.$ (A5)

By using the definition of the temporal mode $A_{2}^{\rm out}(t)$, we have

$m(\tau_{2})=e^{{\cal G}_{2}\tau_{2}}m(0)-i\sqrt{e^{2{\cal
G}_{2}\tau_{2}}-1}A_{2}^{\rm out}(\tau_{2}).$ (A6)

Using the result of Eq. (A2), we get the solution of $A_{2}^{\rm
out}(\tau_{2})$, which is

$A_{2}^{\rm out}(\tau_{2})=-i\sqrt{1-e^{-2{\cal G}_{2}\tau_{2}}}m(0)+e^{-{\cal
G}_{2}\tau_{2}}A_{2}^{\rm in}(\tau_{2}).$ (A7)

Equations (A2) and (A7) are just the solutions of Eq. (22) in the main text.
In the same way, one can derive the solutions of Eq. (40), which lead to an
optomagnonic two-mode squeezed state.

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