# Nonreciprocal Transmission and Entanglement in a cavity-magnomechanical
system

Zhi-Bo Yang1 Jin-Song Liu1 Ai-Dong Zhu1 Hong-Yu Liu1<EMAIL_ADDRESS>Rong-Can Yang2,3<EMAIL_ADDRESS>1Department of Physics, College of
Science, Yanbian University, Yanji, Jilin 133002, China  2Fujian Provincial
Key Laboratory of Quantum Manipulation and New Energy Materials, and College
of Physics and Energy, Fujian Normal University, Fuzhou 350117, China  3Fujian
Provincial Collaborative Innovation Center for Optoelectronic Semiconductors
and Efficient Devices, Xiamen 361005, China

###### Abstract

Quantum entanglement, a key element for quanum information is generated with a
cavity magnomechanical system. It comprises of two microwave cavities, a
magnon mode and a vibrational mode, and the last two elements come from a YIG
sphere trapped in the second cavity. The two microwave cavities are connected
by a superconducting transmission line, resulting in a linear coupling between
them. The magnon mode is driven by a strong microwave field and coupled to
cavity photons via magnetic dipole interaction, and at the same time interacts
with phonons via magnetostrictive interaction. By breaking symmetry of the
configuration, we realize nonreciprocal photon transmission and one-way
bipartite quantum entanglement. By using current experimental parameters for
numerical simulation, it is hoped that our results may reveal a new strategy
to built quantum resources for the realization of noise-tolerant quantum
processors, chiral networks, and so on.

##  I introduction

Although reciprocity is ubiquitous in nature, nonreciprocity promptes diverse
applications such as chiral engineering, invisible sensing, and backaction-
immune information processing 001 ; 002 ; 003 . So far, electromagnetic
nonreciprocal transmission 004 ; 005  has been demonstrated with various
systems ranging from microwave 006 ; 007 ; 008 ; 009 , terahertz 010 , optical
011 ; 012  photons to $x$-rays 013 . As a special type of nonreciprocity which
only allows one-way transmission, unidirectional transmission of light is also
revealed with some hybrid systems, such as atomic gases 014 , nonlinear
devices 011 ; 015 ; 016 ; 017 , moving media 018  and synthetic materials 019
. While in quantum regime, a recent scheme has been proposed, where
nonreciprocal entangled states is prepared by using an optical diode with a
high isolation rate 020 , making it possible to swift a single device between
classical isolator and quantum diode, or to protect quantum resources from
backscattering losses.

Cavity spintronics 021 ; 022 ; 023 ; 024 ; 025 ; 026 ; 027 ; 028 ; 029 ; 030 ;
031 ; 032 ; 033 ; 034 ; 035  is an emerging and rapidly developing
interdisciplinary that studies magnons strongly couple with microwave photons
via magnetic dipole interaction. Such a hybrid system shows great application
prospect in the field of quantum information processing, especially for
quantum transducer 036 ; 037 ; 038 ; 039 ; 040 ; 041 ; 042  and quantum memory
043 . The reason is that magnetic materials have several distinguishing
advantages such as long lifetime, high spin density and easy tunability.
Furthermore, a collective excitation of spins in these materials (called
magnon mode or Kittel mode) in magnetic materials can easily be coupled to a
variety of other types of systems. Thus, cavity spintronics seems to be a
potential candidate to study multiple quantum correlations, etc. By using
cavity spintronics, in this manuscript, we propose a scheme to realize an
optical diode with both quantum and classical characteristics, i.e., the
implementation of both nonreciprocal microwave transmission and nonreciprocal
bipartite quantum entanglement.

In this manuscript, we present a proposal to carry out nonreciprocal microwave
transmission and one-way bipartite entanglement by using an additional
microwave cavity coupled to a cavity-magnon system to break symmetry of
spatial inversion 009 . The two microwave cavities are linked by
superconducting transmission lines 044  and one of them interacts with magnon
mode of a ferrimagnetic yttrium iron garnet (YIG) sphere 045 ; 046 ; 047 ; 048
; 049 ; 021 ; 024 . Simultaneously, the magnon mode is coupled to phonons due
to vibration of YIG sphere induced by the magnetostrictive force 050 ; 051 ;
026 . In addition, magnons are driven by a strong microwave field at the first
red sideband with respect to phonons because entanglement mainly survives with
small thermal phonon occupancy. The (anti-Stokes) process not only realizes
phonon cooling, a prerequisite for observing quantum effects in the system 052
, but also also enhances the effective magnon-phonon coupling in order for the
generation of magnomechanical entanglement. If both microwave cavities are
resonant with the red sideband, then entanglement between the other two
subsystems will be generated though some entanglement is very small. For the
same driving power, different driving direction induces different effective
magnetostrictive interaction, leading the generated subsystem entanglement to
exhibit rare nonreciprocity and unidirectional invisibility. Furthermore, most
of bipartite entanglement is robust against ambient temperature.

Figure 1: (a) A schematic diagram of two-cavity magnomechanic system,
consisting of two microwave cavities and a YIG sphere, which is placed in the
right cavity. The YIG sphere, which is magnetized to saturation by a bias
magnetic field $\textbf{B}_{0}$ aligned along the $z$-direction, is mounted
near the right cavity wall, where the magnetic field of the cavity mode is the
strongest and polarized along $x$-direction to excite the magnon mode in YIG.
The magnon mode is driven by a microwave field along the $y$-direction. Three
magnetic fields are mutually perpendicular ar the site of the YIG sphere. In
addition, the system will exhibit varying degrees of magnomechanical
interaction when the left or right cavity is driven alone. (b) Mode
frequencies and linewidths. The magnon mode (two MW cavity modes) with
frequency $\omega_{m}$ ($\omega_{a}$ and $\omega_{c}$) is driven by a strong
MW field at frequency $\omega_{d}$, and the mechanical motion of frequency
$\omega_{b}$ scatters the driving photons onto two sidebands at
$\omega_{d}\pm\omega_{b}$. If the magnon mode is resonant with the blue (anti-
Stokes) sideband, and two cavity modes are resonant with the red (Stokes)
sideband, all subsystems are prepared in entangled state. See text for more
details.

The manuscript is organized as follows. At first, we present a general model
of the scheme, and then solve the system dynamics by means of the standard
Langevin formalism with linearization treatment. Next, we illustrate
nonreciprocity of microwave transmission and bipartite entanglement in the
stationary state. Finally, we show how to measure generated entanglement and
analyze the validity of our model.

##  II Model and equation of motion

As schematically shown in Fig. 1 (a), we study a cavity magnomechanical system
which consists of two coupled microwave cavities and one of them coupled to a
YIG sphere. The sphere is uniformly magnetized to saturation by a bias
magnetic field with $\textbf{B}_{0}=B_{0}\textbf{e}_{z}$, where $B_{0}$ and
$\textbf{e}_{z}$ represent magnetic amplitude and the unit vector in $z$
direction, respectively 023 . At the same time, the YIG sphere is also
directly driven by a microwave field with driving strength $E_{m}$ and
frequency $\omega_{d}$. In addition, either the first cavity or the second one
is driven by a microwave light beam with driving strength $E_{a}$ or $E_{c}$
with the same frequency $\omega_{d}$. The two cavities are linked by a
superconducting transmission line 044  and the magnon mode couples to the
second cavity mode and a vibrational mode 021  via the collective magnetic-
dipole interaction and magnetostrictive interaction, respectively. The
magnetostatic mode with finite wave number has a distinct frequency different
from the Kittel mode so that the selective excitation may be implemented
through the driving field wavelength and cavity mode selection. When all of
the driving fields are included, the total Hamiltonian of the system can be
written as follows

$\displaystyle\mathcal{H}/\hbar$ $\displaystyle=$
$\displaystyle\omega_{a}\hat{a}^{{\dagger}}\hat{a}+\omega_{c}\hat{c}^{{\dagger}}\hat{c}+\omega_{m}\hat{m}^{{\dagger}}\hat{m}+\omega_{b}\hat{b}^{{\dagger}}\hat{b}+g_{ac}(\hat{a}^{{\dagger}}\hat{c}+\hat{c}^{{\dagger}}\hat{a})$
(1)
$\displaystyle+g_{cm}(\hat{c}^{{\dagger}}\hat{m}+\hat{m}^{{\dagger}}\hat{c})+g_{mb}\hat{m}^{{\dagger}}\hat{m}(\hat{b}^{{\dagger}}+\hat{b})$
$\displaystyle+iE_{a}(\hat{a}^{{\dagger}}e^{-i\omega_{d}t}-\hat{a}e^{i\omega_{d}t})+iE_{c}(\hat{c}^{{\dagger}}e^{-i\omega_{d}t}-\hat{c}e^{i\omega_{d}t})$
$\displaystyle+iE_{m}(\hat{m}^{{\dagger}}e^{-i\omega_{d}t}-\hat{m}e^{i\omega_{d}t}).$

Here, $\hat{a}$, $\hat{c}$, $\hat{m}$ and $\hat{b}$ ($\hat{a}^{{\dagger}}$,
$\hat{c}^{{\dagger}}$, $\hat{m}^{{\dagger}}$ and $\hat{b}^{{\dagger}}$) are
individually the annihilation (creation) operators of corresponding cavities,
magnons and phonons with resonance frequency $\omega_{a}$, $\omega_{c}$,
$\omega_{m}$ and $\omega_{b}$. The magnon frequency is determined by the
external bias magnetic field $B_{0}$, i.e., $\omega_{m}=\gamma B_{0}$ with
$\gamma/2\pi=2.8$ MHz/Oe being the gyromagnetic ratio. $g_{ac}$ is the photon
coupling rate 053  and $g_{cm}$ represents the linear photon-magnon coupling
strength which can be estimated by measuring the reflection spectrum of YIG
sphere in right cavity and can be adjusted by varying the direction of the
bias field or the position of the YIG sphere inside the right cavity 050 . The
single-magnon magnomechanical coupling rate $g_{mb}$ is typically small, but
the magnomechanical interaction can be enhanced if magnons are strongly driven
021 ; 022 ; 054 ; 055 . The amplitude for each driving fields is
$E_{i}=\sqrt{\kappa_{i}}\varepsilon_{i}$ with the effective strength
$\varepsilon_{i}=\sqrt{P_{i}/\hbar\omega_{d}}$ with the corresponding driving
power and driving frequency being $P_{i}$ and $\omega_{d}$, respectively 025 .
$\kappa_{a/c}$ and $\kappa_{m/b}$ separately represent the total loss rate of
the first/second cavity and the magnon/phonon mode.

In the frame rotating at the driving frequency $\omega_{d}$, the quantum
Langevin equations (QLEs) describing the system are given by

$\displaystyle\dot{\hat{a}}$ $\displaystyle=$
$\displaystyle-(i\Delta_{a}+\kappa_{a})\hat{a}-ig_{ac}\hat{c}+E_{a}+\sqrt{2\kappa_{a}}\hat{a}^{in},$
$\displaystyle\dot{\hat{c}}$ $\displaystyle=$
$\displaystyle-(i\Delta_{c}+\kappa_{c})\hat{c}-ig_{ac}\hat{a}-ig_{cm}\hat{m}+E_{c}+\sqrt{2\kappa_{c}}\hat{c}^{in},$
$\displaystyle\dot{\hat{m}}$ $\displaystyle=$
$\displaystyle-(i\Delta_{m}+\kappa_{m})\hat{m}-ig_{cm}\hat{c}-ig_{mb}\hat{m}(\hat{b}^{{\dagger}}+\hat{b})+E_{m}$
$\displaystyle+\sqrt{2\kappa_{m}}\hat{m}^{in},$ $\displaystyle\dot{\hat{b}}$
$\displaystyle=$
$\displaystyle-(i\omega_{b}+\kappa_{b})\hat{b}-ig_{mb}\hat{m}^{{\dagger}}\hat{m}+\sqrt{2\kappa_{b}}\hat{b}^{in},$
(2)

where $\Delta_{j}=\omega_{j}-\omega_{d}$ and $\hat{a}^{in}$, $\hat{c}^{in}$,
$\hat{m}^{in}$, $\hat{b}^{in}$ are input noise operators with zero mean value
acting on the cavities, magnon and mechanical modes, respectively, which are
characterized by the following correlation functions:
$\langle\hat{a}^{in}(t)\hat{a}^{in{\dagger}}(t^{{}^{\prime}})\rangle=\langle\hat{c}^{in}(t)\hat{c}^{in{\dagger}}(t^{{}^{\prime}})\rangle=\langle\hat{m}^{in}(t)\hat{m}^{in{\dagger}}(t^{{}^{\prime}})\rangle=\delta(t-t^{{}^{\prime}})$,
$\langle\hat{b}^{in}(t)\hat{b}^{in{\dagger}}(t^{{}^{\prime}})\rangle=(n_{b}+1)\delta(t-t^{{}^{\prime}})$,
and
$\langle\hat{b}^{in{\dagger}}(t)\hat{b}^{in}(t^{{}^{\prime}})\rangle=n_{b}\delta(t-t^{{}^{\prime}})$
where the equilibrium mean thermal phonon numbers
$n_{b}=[\exp(\frac{\hbar\omega_{b}}{k_{B}T})-1]^{-1}$ with $k_{B}$ the
Boltzmann constant and $T$ the ambient temperature.

Because the magnon mode and microwave photons are strongly driven, they all
have large amplitude, i.e., $\arrowvert\langle a\rangle\arrowvert$,
$\arrowvert\langle c\rangle\arrowvert$, $\arrowvert\langle m\rangle\arrowvert$
$\gg 1$, which allows us to linearize the dynamics of the system around the
steady-state values by writing any a operator as $\hat{o}=\langle
o\rangle+\delta o$ ($o=a,c,m,b$) and neglecting second order fluctuation
terms. Then, we obtain a set of differential equations for the mean values:

$\displaystyle\dot{\langle a\rangle}$ $\displaystyle=$
$\displaystyle-(i\Delta_{a}+\kappa_{a})\langle a\rangle-ig_{ac}\langle
c\rangle+E_{a},$ $\displaystyle\dot{\langle c\rangle}$ $\displaystyle=$
$\displaystyle-(i\Delta_{c}+\kappa_{c})\langle c\rangle-ig_{ac}\langle
a\rangle-ig_{cm}\langle m\rangle+E_{c},$ $\displaystyle\dot{\langle m\rangle}$
$\displaystyle=$ $\displaystyle-(i\tilde{\Delta}_{m}+\kappa_{m})\langle
m\rangle-ig_{cm}\langle c\rangle+E_{m},$ $\displaystyle\dot{\langle b\rangle}$
$\displaystyle=$ $\displaystyle-(i\omega_{b}+\kappa_{b})\langle b\rangle-
ig_{mb}\arrowvert\langle m\rangle\arrowvert^{2},$ (3)

and the linearized QLEs for the quantum fluctuations:

$\displaystyle\delta\dot{a}$ $\displaystyle=$
$\displaystyle-(i\Delta_{a}+\kappa_{a})\delta a-ig_{ac}\delta
c+\sqrt{2\kappa_{a}}\hat{a}^{in},$ $\displaystyle\delta\dot{c}$
$\displaystyle=$ $\displaystyle-(i\Delta_{c}+\kappa_{c})\delta c-ig_{ac}\delta
a-ig_{cm}\delta m+\sqrt{2\kappa_{c}}\hat{c}^{in},$
$\displaystyle\delta\dot{m}$ $\displaystyle=$
$\displaystyle-(i\tilde{\Delta}_{m}+\kappa_{m})\delta m-ig_{cm}\delta
c-\frac{1}{2}G_{mb}(\delta b^{{\dagger}}+\delta b)$
$\displaystyle+\sqrt{2\kappa_{m}}\hat{m}^{in},$ $\displaystyle\delta\dot{b}$
$\displaystyle=$ $\displaystyle-(i\omega_{b}+\kappa_{b})\delta
b-\frac{1}{2}G_{mb}(\delta m^{{\dagger}}-\delta
m)+\sqrt{2\kappa_{b}}\hat{b}^{in},$

where $\tilde{\Delta}_{m}=\Delta_{m}+g_{mb}(\langle b\rangle+\langle
b\rangle^{*})$ is the effective detuning of the magnon mode including the
frequency shift caused by the magnetostrictive interaction.
$G_{mb}=i2g_{mb}\langle m\rangle$ is the effective magnomechanical coupling
rate. If we consider the time to be $t\to\infty$ and the detunings to satisfy
$\arrowvert\Delta_{a}\arrowvert$, $\arrowvert\Delta_{c}\arrowvert$,
$\arrowvert\tilde{\Delta}_{m}\arrowvert$ $\gg$ $\kappa_{a}$, $\kappa_{c}$,
$\kappa_{m}$, then $\langle a\rangle$, $\langle c\rangle$, $\langle m\rangle$
and $\langle b\rangle$ can be given by

$\displaystyle\langle a\rangle$ $\displaystyle\simeq$ $\displaystyle
i\frac{E_{m}g_{ac}g_{cm}-E_{a}g_{cm}^{2}-E_{c}g_{ac}\tilde{\Delta}_{m}+E_{a}\Delta_{c}\tilde{\Delta}_{m}}{g_{cm}^{2}\Delta_{a}+g_{ac}^{2}\tilde{\Delta}_{m}-\Delta_{a}\Delta_{c}\tilde{\Delta}_{m}},$
$\displaystyle\langle c\rangle$ $\displaystyle\simeq$
$\displaystyle-i\frac{E_{m}g_{cm}\Delta_{a}+E_{a}g_{ac}\tilde{\Delta}_{m}-E_{c}\Delta_{a}\tilde{\Delta}_{m}}{g_{cm}^{2}\Delta_{a}+g_{ac}^{2}\tilde{\Delta}_{m}-\Delta_{a}\Delta_{c}\tilde{\Delta}_{m}},$
$\displaystyle\langle m\rangle$ $\displaystyle\simeq$
$\displaystyle-i\frac{E_{m}g_{ac}^{2}-E_{a}g_{ac}g_{cm}+E_{c}g_{cm}\Delta_{a}-E_{m}\Delta_{a}\Delta_{c}}{g_{cm}^{2}\Delta_{a}+g_{ac}^{2}\tilde{\Delta}_{m}-\Delta_{a}\Delta_{c}\tilde{\Delta}_{m}},$
$\displaystyle\langle b\rangle$ $\displaystyle=$ $\displaystyle-
ig_{mb}\arrowvert\langle m\rangle\arrowvert^{2}/(i\omega_{b}+\kappa_{b}).$

In what follows, we first show that the effect of the related parameters on
the nonreciprocal transmission of a microwave field with a forward and
backward driving-field input by solving the equations numerically, and then we
show that the classical nonreciprocity can also be used to prepare
nonreciprocal subsystem entangled states.

##  III Nonreciprocal transmission

In order to study the nonreciprocity of cavity output field, we let the first
cavity mode driven ($E_{a}\neq 0\quad$ & $\quad E_{c}=0$) or the second cavity
mode driven ($E_{a}=0\quad$ & $\quad E_{c}\neq 0$) by a classical field, and
measure the output field of the other cavity 025 . For the sake of concision,
we denote the first (second) case as the driving-field input from the forward
(backward) direction in the following passages. Thus, the transmission
coefficients for the two cases are separately defined as
$\mathcal{T}_{12}=\arrowvert
c^{out}/\varepsilon_{a}\arrowvert=\arrowvert\sqrt{\kappa_{c}}\langle
c\rangle/\varepsilon_{a}\arrowvert$ and $\mathcal{T}_{21}=\arrowvert
a^{out}/\varepsilon_{c}\arrowvert=\arrowvert\sqrt{\kappa_{a}}\langle
a\rangle/\varepsilon_{c}\arrowvert$ with the average number of the first
(second) cavity $\arrowvert\langle a(c)\rangle\arrowvert$ calculated with Eq.
(II). In addition, in order to depict the nonreciprocality, we define the
isolation $\mathcal{T}_{iso}=20\times
log_{10}\arrowvert\mathcal{T}_{12}/\mathcal{T}_{21}\arrowvert$ which is given
in decibels (dB) 009 ; 025 . For the sake of simplicity, we set the power of
the microwave source focused on the first or second cavity to be the same,
i.e. $P_{a}=P_{c}=P$, and the two cavities to be identical, i.e.
$\kappa_{a}=\kappa_{c}=\kappa$. The transmission coefficient
$\mathcal{T}_{12/21}$ and the isolation $\mathcal{T}_{iso}$ as a function of
the driving power $P$ are plotted in Fig. 2 (a) and (c) with
$g_{ac}/\omega_{b}=1$, and (b) and (d) with $g_{ac}/\omega_{b}=0.32$. The
other parameters are chosen as $\omega_{a}/2\pi=10$ GHz, $\omega_{b}/2\pi=10$
MHz, $\tilde{\Delta}_{m}=0.9\omega_{b}$, $\Delta_{c}=\Delta_{a}=-\omega_{b}$,
$\kappa/2\pi=\kappa_{m}/2\pi=1$ MHz, and $g_{cm}/2\pi=3.2$ MHz 054 ; 023 .

From Eq. (II), we find that when $g_{ac}=\omega_{b}$ the same transmission
coefficient is observed for the different directions (forward or backward
injection) of the driving-field input. We regard $g_{ac}=\omega_{b}$ as a
impedance-matching condition 053  for the same microwave transmission. From
Fig. 2 (a) with $g_{ac}/\omega_{b}=1$, we can clearly see that the two
transmission coefficients ($\mathcal{T}_{12}$ and $\mathcal{T}_{21}$) are
always equal. However, if we decrease the coupling rate $g_{ac}$ between the
two cavities, then the two coefficients ($\mathcal{T}_{12}$ and
$\mathcal{T}_{21}$) will be distinctly different when $P$ is small. For
example, $\mathcal{T}_{12}\simeq 0.08,0$ and $\mathcal{T}_{21}\simeq 0,0.02$
when $P\simeq 10,110$ mW (see Fig. 2 (b)). While the increase of $P$ will lead
$\mathcal{T}_{12}$ and $\mathcal{T}_{21}$ to be closer and then to keep some
difference. Therefore, if the impedance-matching condition is broken, the
transmission for both two directions will be distinct. This is a typical
nonreciprocal phenomenon, which corresponds to the classical nonreciprocal
microwave transmission 009 . In addition, the degree of the nonreciprocal
transmission of a microwave field can be described by the isolation as
follows: $\mathcal{T}_{iso}=20\times
log_{10}\arrowvert\mathcal{T}_{12}/\mathcal{T}_{21}\arrowvert$ given in
decibels (dB). The calculation results for $\mathcal{T}_{iso}$, plotted as a
function of driving power, is shown in Fig. 2 (c) and (d). Fig. 2 (c) shows
that when the impedance-matching condition is met, the microwave transmission
coefficients in the two directions are almost the same, i.e.,
$\mathcal{T}_{iso}=0$. However, Fig. 2 (d) gives us a clearer perspective for
the differences between the transmission coefficients $\mathcal{T}_{12}$ and
$\mathcal{T}_{21}$. At this time, the impedance-matching condition is broken,
and the microwave transmission in the two directions has a larger transmission
isolation ratio $\mathcal{T}_{iso}>70$ dB. It is worth noting that when $P=0$,
$\mathcal{T}_{iso}\not=0$ in Fig. 2 (d), this is because the magnons mode is
always driven by a microwave source with driving power $P=94.5$ mW to maintain
a certain magnitude of magnomechanical coupling.

Figure 2: (a) (b) Transmission coefficients $\mathcal{T}_{12}$ and
$\mathcal{T}_{21}$ and (c) (d) transmission isolation ratio $T_{iso}$ versus
driving power $P$ for (a) (c) $g_{ac}/\omega_{b}=1$ and (b) (d)
$g_{ac}/\omega_{b}=0.32$, where $P_{m}=94.5$ mW. Other parameters are
$\omega_{a}/2\pi=10$ GHz, $\omega_{b}/2\pi=10$ MHz,
$\tilde{\Delta}_{m}=0.9\omega_{b}$, $\Delta_{c}=\Delta_{a}=-\omega_{b}$,
$\kappa/2\pi=\kappa_{m}/2\pi=1$ MHz, and $g_{cm}/2\pi=3.2$ MHz which are
mostly based on the latest experimental parameters 054 ; 023 .

##  IV Definition of entanglement

We thereby obtain the linearized QLEs for the quadratures $\delta X_{o},\delta
Y_{o}$ defined as $\delta X_{o}=(\delta o+\delta
o^{{\dagger}})/\sqrt{2},\delta Y_{o}=(\delta o-\delta
o^{{\dagger}})/i\sqrt{2}$, can be written as $\dot{\sigma}(t)=A\sigma(t)+n(t)$
with $\sigma(t)=[\delta X_{a}(t),\delta Y_{a}(t),\delta X_{c}(t),\delta
Y_{c}(t),\delta X_{m}(t),\delta Y_{m}(t),\delta X_{b}(t),\\\ \delta
Y_{b}(t)]^{T}$ and $n(t)=[\sqrt{2\kappa_{a}}\delta
X_{a}^{in}(t),\sqrt{2\kappa_{a}}\delta Y_{a}^{in}(t),\sqrt{2\kappa_{c}}\\\
\delta X_{c}^{in}(t),\sqrt{2\kappa_{c}}\delta
Y_{c}^{in}(t),\sqrt{2\kappa_{m}}\delta X_{m}^{in}(t),\sqrt{2\kappa_{m}}\delta
Y_{m}^{in}(t),\\\ \sqrt{2\kappa_{b}}\delta
X_{b}^{in}(t),\sqrt{2\kappa_{b}}\delta Y_{b}^{in}(t)]^{T}$ being the vectors
for quantum fluctuations and noise, respectively. In addition, the drift
matrix $A$ reads

$\displaystyle A=\begin{bmatrix}-\kappa_{a}&\Delta_{a}&0&g_{ac}&0&0&0&0\\\
-\Delta_{a}&-\kappa_{a}&-g_{ac}&0&0&0&0&0\\\
0&g_{ac}&-\kappa_{c}&\Delta_{c}&0&g_{cm}&0&0\\\
-g_{ac}&0&-\Delta_{c}&-\kappa_{c}&-g_{cm}&0&0&0\\\
0&0&0&g_{cm}&-\kappa_{m}&\tilde{\Delta}_{m}&-G_{mb}&0\\\
0&0&-g_{cm}&0&-\tilde{\Delta}_{m}&-\kappa_{m}&0&0\\\
0&0&0&0&0&0&-\kappa_{b}&\omega_{b}\\\
0&0&0&0&0&G_{mb}&\omega_{b}&-\kappa_{b}\\\ \end{bmatrix}.$

In this case, when the forward input direction of the driving field is taken
into account (i.e., $E_{a}\not=0$ and $E_{c}=0$), magnomechanical coupling can
be written as follows

$\displaystyle G_{mb,12}$ $\displaystyle=$
$\displaystyle\frac{2g_{mb}(E_{m}g_{ac}^{2}-E_{a}g_{ac}g_{cm}-E_{m}\Delta_{a}\Delta_{c})}{g_{cm}^{2}\Delta_{a}+g_{ac}^{2}\tilde{\Delta}_{m}-\Delta_{a}\Delta_{c}\tilde{\Delta}_{m}}.$
(7)

However, when the backward input direction of the driving field is taken into
account (i.e., $E_{a}=0$ and $E_{c}\not=0$), magnomechanical coupling changes
to

$\displaystyle G_{mb,21}$ $\displaystyle=$
$\displaystyle\frac{2g_{mb}(E_{m}g_{ac}^{2}+E_{c}g_{cm}\Delta_{a}-E_{m}\Delta_{a}\Delta_{c})}{g_{cm}^{2}\Delta_{a}+g_{ac}^{2}\tilde{\Delta}_{m}-\Delta_{a}\Delta_{c}\tilde{\Delta}_{m}}.$
(8)

Since we are using linearized QLEs, the Gaussian nature of the input states
will be preserved during the time evolution for the system. So, the quantum
fluctuation is in a continuous four-mode Gaussian state which can be
completely characterized by a $8\times 8$ covariance matrix (CM) $V$ in the
phase space
$2V_{ij}(t,t^{{}^{\prime}})=\langle\sigma_{i}(t)\sigma_{j}(t^{{}^{\prime}})+\sigma_{j}(t^{{}^{\prime}})\sigma_{i}(t)\rangle$,
($i,j=1,\cdots,8$). Then the vector can be obtained straightforwardly by
solving the Lyapunov equation

$\displaystyle AV+VA^{T}=-D$ (9)

with $D=$ diag
$[\kappa_{a},\kappa_{a},\kappa_{c},\kappa_{c},\kappa_{m},\kappa_{m},(2n_{b}+1)\kappa_{b},(2n_{b}+1)\kappa_{b}]$
defind through $2D_{ij}\delta(t-t^{{}^{\prime}})=\langle
n_{i}(t)n_{j}(t^{{}^{\prime}})+n_{j}(t^{{}^{\prime}})n_{i}(t)\rangle$. In this
manuscript, we use logarithmic negativity 058 ; 059  to quantify the degree of
the quantum entanglement for the four-mode Gaussian state, which is defined as
$E_{N}\equiv\max[0,-2\ln\nu^{-}]$, where $\nu^{-}=\min$ eig
$\arrowvert\oplus_{j=1}^{2}(-\sigma_{y})PVP\arrowvert$ with $\sigma_{y}$ and
$P=\sigma_{z}\oplus 1$ are respectively $y$-Pauli matrix and the matrix that
realizes partial transposition at the CM level 058 ; 060 .

Figure 3: (a) $E_{ac}$, (b) $E_{cm}$, (c) $E_{mb}$ and (d) $E_{ab}$ versus
detunings $\Delta_{a}$ and $\Delta_{c}$. We take $G_{mb}/2\pi=2.5$ MHz,
$\kappa_{b}/2\pi=100$ Hz, $T=20$ mK and the other parameters are same as Fig.
2.

##  V Nonreciprocal entanglement

The foremost task of studying entanglement properties between any two
subsystems in such a hybrid system is to find the optimal effective
interaction among modes, i.e., to find optimal frequency detuning that can
generate subsystem entanglement 021 . In Fig. 3, we show four types of
subsystem entanglement ($E_{ac}$, $E_{cm}$, $E_{mb}$ and $E_{ab}$) as the
function of cavity detunings $\Delta_{a}$ and $\Delta_{c}$, where $E_{ac}$,
$E_{cm}$, $E_{mb}$ and $E_{ab}$ denote the cavity-cavity entanglement, cavity-
magnon entanglement, magnon-phonon entanglement, and cavity-phonon distant
entanglement, respectively. In addition, we choose $G_{mb}/2\pi=2.5$ MHz,
$\kappa_{a}=\kappa_{c}=\kappa_{m}$, $\kappa_{b}/2\pi=100$ Hz, $T=20$ mK and
the other parameters are chosen as the same as that in Fig. 2. Furthermore, we
also set $\tilde{\Delta}_{m}\simeq\omega_{b}$ which imply that magnon mode is
in the blue sideband with respect to the first cavity mode, which corresponds
to the anti-Stokes process, i.e., significantly cooling the phonon mode. Thus,
the elimination of the main obstacle for observing entanglement is obtained
021 . It is noted that all results are satisfying with the condition of the
steady state guaranteed by the negative eigenvalues (real parts) of the drift
matrix $A$. From Fig. 3, it is shown that a parameter regime exists, i.e.,
$\Delta_{a}=\Delta_{c}=-\omega_{b}$, where the entanglement within any two
subsystems occurs (see Fig. 1 (b)). This is similar to the realization of
entanglement between magnon modes in a magnomechanic system with two YIG
spheres proposed in Ref. 061 . In order to obtain the entanglement between any
two subsystems and keep the system stable at the same time, the three coupling
rates $g_{ac}$, $g_{cm}$ and $G_{mb}$ should be on the same order of magnitude
and chosen as a separate and moderate value. Based on it, we choose
$g_{ac}g_{cm}G_{mb}\ll\arrowvert\Delta_{a}\Delta_{c}\tilde{\Delta}_{m}\arrowvert\simeq\omega_{b}^{3}$
and $G_{mb}=2.5$ MHz which corresponds to $E_{m}\simeq
G_{mb}\omega_{b}^{3}/2g_{mb}(\omega_{b}^{2}-g_{ac}^{2})\simeq 4.2\times
10^{13}$ Hz and $P_{m}\simeq 1.85$ mW, when $E_{a}=E_{c}=0$ in Fig. 3. All
entanglement in the system comes from the magnetostrictive interaction between
the magnon and the phonon 021 . Next, we will show the realization of
nonreciprocal subsystem entanglement by controlling the classical
nonreciprocal transmission, i.e., controlling the effective magnomechanical
interaction.

Figure 4: (a) $E_{mb}$, (b) $E_{ab}$, (c) $E_{mb,iso}$, and (d) $E_{ab,iso}$
versus driving power $P$. We take $g_{mb}/2\pi=0.3$ Hz, $\kappa_{b}/2\pi=100$
Hz, $T=20$ mK and the other parameters are same as Fig. 2.

For the sake of brevity, we mainly focus on the entanglement $E_{ab}$ and
$E_{mb}$ due to they being larger than the other types of entanglement. In
Fig. 4 (a) (b), we show $E_{mb}$ and $E_{ab}$ as the function of driving power
$P$ on the cavity $a$ or $c$, where $E_{ij,12}$ ($E_{ij,21}$) represents the
entanglement between mode $i$ and mode $j$ when the direction of driving
microwave source is forward (backward). Additionally, $g_{mb}/2\pi=0.3$ Hz,
$\kappa_{a}=\kappa_{c}=\kappa_{m}$ and $T=20$ mK are set and the other
parameters are chosen as the same as that in Fig. 2. Due to the existence of
magnon-phonon nonlinear coupling (magnetostrictive force), the entanglement of
the magnomechanical subsystem is generated. And because there is a direct or
indirect linear state-swap interaction between the two microwave cavities and
the magnon mode, the resulting magnomechanical entanglement is distributed to
other subsystems. Therefore, controlling the optimal effective magnomechanical
interaction will be an effective operation to control whether entanglement
exists in each subsystem. From Fig. 4 (a) we can see that the magnomechanical
subsystem entanglement in the two directions shows very different trends with
the increase of the driving power. Therefore, entanglement of other subsystems
will be affected accordingly (see Fig. 4 (d)). The results in Fig. 4 (a) (b)
demonstrate that the entanglement of multiple subsystems in a hybrid system
can be prepared in a highly asymmetric way. This is in good agreement with our
previous expectations. This is a clear signature of quantum nonreciprocity,
which is fundamentally different from that in classical devices 062 ; 063
showing only nonreciprocal transmission rates.

Figure 5: (a) $E_{mb}$ and (b) $E_{ab}$ versus ambient temperature $T$. We
take $P=0.5$ W, $g_{mb}/2\pi=0.3$ Hz, $\kappa_{b}/2\pi=100$ Hz and the other
parameters are same as Fig. 2.

Next, the difference between subsystem entanglement with forward driving
direction and backward driving direction is extracted in the decibel scale
(defined as $20\times log_{10}\arrowvert E_{ij,12}/E_{ij,21}\arrowvert$,
similar to $\mathcal{T}_{iso}$ 003 ), and we take its value as the
entanglement isolation ratio $E_{ij,iso}$ (units of dB). For the case of
reciprocal subsystem entanglement, we have $E_{ij,12}/E_{ij,21}=1$ and
$E_{ij,iso}=0$. A nonzero $E_{ij,iso}$ presents nonreciprocal entanglement and
the greater the value of $E_{ij,iso}$, the higher is the degree of the
nonreciprocal entanglement. The calculation results for $E_{ij,iso}$, plotted
as a function of the driving power $P$, are shown in Fig. 4 (c) (d), which
gives us a clearer perspective for the differences between the entanglements
$E_{ij,12}$ and $E_{ij,21}$. Notice the cutoff in Fig. 4 (d). The position of
the cutoff corresponds to the position of $E_{ab,21}=0$ in Fig. 4 (b), which
shows the unidirectional invisibility of subsystem entanglement. In the
multilayer microwave integrated quantum circuit, we can further design
superconducting transmission lines and interconnects based on these existing
technologies to provide the large range of necessary couplings and to minimize
any parasitic losses 044 . In fact, we can find from Eqs. (7–8) that when
$g_{ac}=\omega_{b}$, the system reaches the impedance matching condition 025 ;
053 , i.e., $G_{mb,12}=G_{mb,21}$. This realizes the switch from nonreciprocal
to reciprocal of subsystem entangled state and shows the potential advantage
of our solution as a tunable quantum diode.

Fig. 5, $E_{mb}$ and $E_{ab}$ as a function of ambient temperature $T$ for
driving power $P=1$ W, shows that the generated subsystem entanglements is
robust to ambient temperature and survives up to $\sim 100$ mK, below which
the average phonon number is always smaller than 1, showing that mechanical
cooling is, thus, a precondition for observing quantum entanglement in the
system 021 . Compared with the scheme that a strong squeezed vacuum field
proposed in Ref. 064  is used to generate entanglement between magnon modes,
in our scheme due to the inherent low frequency of phonon modes, this
robustness is generally weak.

##  VI Discussion and conclusion

Lastly, we discuss how to detect the entanglement and verify the effectiveness
of two-cavity magnomechanic system. The generated subsystem entanglements can
be detected by measuring the CM of two cavity output fields 021 . Such
measurement in the microwave domain has been realized in the experiments 065 ;
066 . In addition, for a 0.5-mm-diameter YIG sphere, the number of spins
$N\simeq 2.8\times 10^{17}$, and $P_{m}=189$ mW corresponds to $E_{m}\simeq
3\times 10^{14}$ Hz, and $\arrowvert\langle m\rangle\arrowvert\simeq 5.9\times
10^{6}$, leading to $\langle m^{{\dagger}}m\rangle\simeq 3.5\times 10^{13}\ll
5N=1.4\times 10^{18}$ which is well fulfilled. It is worth noting that two
cavity modes is resonant with the red-sideband
($\Delta_{a}=\Delta_{c}=-\omega_{b}$) results in a higher magnon excitation
number in the stable state in the presence of $E_{a}$ or $E_{c}$, where the
magnon number has a simpler form $\langle
m^{{\dagger}}m\rangle\simeq[E_{m}(\omega_{b}^{2}-g_{ac}^{2})+E_{a}g_{ac}g_{cm}]^{2}/\omega_{b}^{6}$
or $\langle
m^{{\dagger}}m\rangle\simeq[E_{m}(\omega_{b}^{2}-g_{ac}^{2})+E_{c}\omega_{b}g_{cm}]^{2}/\omega_{b}^{6}$.
Therefore, under the premise that the magnon mode is continuously driven by
the classical microwave field with driving power $P_{m}=94.5$ mW, $P=1$ W
corresponds to $\langle m^{{\dagger}}m\rangle_{12}\simeq 1.4\times 10^{15}\ll
5N$ ($E_{a}\not=0$, $E_{c}=0$) and $\langle m^{{\dagger}}m\rangle_{21}\simeq
3.47\times 10^{15}\ll 5N$ ($E_{c}\not=0$, $E_{a}=0$), respectively. In order
to keep the Kerr effect negligible, $K\arrowvert\langle
m\rangle\arrowvert^{3}\ll\sum_{i=a,c,m}E_{i}$ must hold 021 . Kerr coefficient
$K$ is inversely proportional to the volume of the sphere. In this manuscript,
we use a 5-mm-diameter YIG sphere, $K/2\pi\simeq 8\times 10^{-10}$ Hz which
corresponds to $K\arrowvert\langle m\rangle\arrowvert^{3}\simeq 4.2\times
10^{13}$ Hz $\ll E_{m}+E_{a}\simeq 1.3\times 10^{15}$ ($P_{m}=189$ mW, $P=2$
W, and $E_{c}=0$) and $K\arrowvert\langle m\rangle\arrowvert^{3}\simeq
1.64\times 10^{14}$ Hz $\ll E_{m}+E_{c}\simeq 1.3\times 10^{15}$ ($P_{m}=189$
mW, $P=2$ W, and $E_{a}=0$). This implying that the nonlinear effects are
negligible and the linearization treatment of the model is a good
approximation.

In summary, we show how to use an asymmetric cavity magnomechanic system to
produce classical nonreciprocal transmission, and extend this nonreciprocity
to quantum states, thereby generating nonreciprocal subsystem entangled
states, rather than by means of nonreciprocal devices 062 . The introduction
of an additional cavity mode successfully breaks the symmetry of spatial
inversion. Our work opens up a range of exciting opportunities for quantum
information processing, networking and metrology by exploiting the power of
quantum nonreciprocity. The ability to manipulate quantum states or
nonclassical correlations in a nonreciprocal way sheds new lights on chiral
quantum engineering and can stimulate more works on achieving and operating
quantum nonreciprocal devices, such as directional quantum squeezing 067 ; 068
, backaction-immune quantum sensing 069 ; 070 , and quantum chiral coupling of
cavity optomechanics devices to superconducting qubits or atomic spins 021 ;
071 ; 072 .

##  VII Acknowledgments

This work is supported by the Science and Technology project of Jilin
Provincial Education Department of China during the 13th Five-Year Plan Period
(Grant No. JJKH20200510KJ) and the Fujian Natural Science Foundation (Grant
No. 2018J01661 and No. 2019J01431).

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