# Enhanced bipartite entanglement and Gaussian quantum steering of squeezed
magnon modes

Shaik Ahmed  School of Technology, Woxsen University, Hyderabad, Telangana
-502345, India  M. Amazioug  LPTHE-Department of Physics, Faculty of sciences,
Ibn Zohr University, Agadir, Morocco  Jia-Xin Peng  State Key Laboratory of
Precision Spectroscopy, Quantum Institute for Light and Atoms, Department of
Physics, East China Normal University, Shanghai 200062, China  S. K.
Singh111Corresponding Author<EMAIL_ADDRESS>Graphene and Advanced
2D Materials Research Group, Sunway University, Malaysia

###### Abstract

We theoretically investigate a scheme to entangle two squeezed magnon modes in
a double cavity-magnon system, where both cavities are driven by a two-mode
squeezed vacuum microwave field. Each cavity contains an optical parametric
amplifier as well as a macroscopic yttrium iron garnet (YIG) sphere placed
near the maximum bias magnetic fields such that this leads to the excitation
of the relevant magnon mode and its coupling with the corresponding cavity
mode. We have obtained optimal parameter regimes for achieving the strong
magnon-magnon entanglement and also studied the effectiveness of this scheme
towards the mismatch of both the cavity-magnon couplings and decay parameters.
We have also explored the entanglement transfer efficiency including Gaussian
quantum steering in our proposed system.

##  I Introduction

Quantum entanglement [1] and Gaussian quantum steering [2, 3] are two major
important resources in the field of quantum computing [4], quantum
cryptography [5] and quantum teleportation [6] including quantum information
processing [7]. Many microscopic as well as macroscopic quantum systems have
been proposed over the past decades for the study of quantum entanglement and
other nonclassical quantum correlations in superconducting qubits [8], atomic
ensembles [9], cavity optomechanics [10, 11, 12, 13, 14, 15, 16] and cavity
magnomechanical (CMM) systems [17, 18, 19, 20, 21, 22] which paves the way for
advancements in present era of quantum technology. In CMM systems, the magnons
defined as the collective excitation of a large number of spins in
ferrimagnetic materials play very important role in the study of light-matter
interactions due to their tunability, low damping, high spin density [23, 24]
as well the strong coupling with the microwave photons [25, 26, 27, 28].
Moreover, other important mascroscopic quantum phenomena such as magnon-
induced effects [30], tunable magnomechanically induced transparency and
absorption [31, 32], slow light [33], four-wave mixing [34], squeezed states
[35, 36, 37, 38], nonclassical quantum correlations [39, 40], microwave-to-
optical carrier conversion [41] including quantum sensing [42, 43] also
successfully investigated in cavity magnomechanical systems.
To quantify the bipartite entanglement between the magnon and microwave photon
in CMM systems, we use a very well-known witness of bipartite entanglement
defined as the logarithmic negativity [44]. A recent theoretical work given in
[45] explored the logarithmic negativity between two magnon modes where the
optimal conditions for achieving the strong magnon-magnon entanglement involve
the resonant coupling between the microwave cavity with both the magnon modes
whereas in case of two microwave cavities given in [46, 47, 48], it is found
that the detuning of the cavity and magnon mode significantly affects the
bipartite entanglement. These research works also found the presence of both
one-way and two-way Gaussian quantum steering. So, all these studies
demonstrate that the bipartite entanglement and quantum steering in CMM
systems can be significantly controlled through the various physical
parameters. All these recent progress also broaden our understanding of
quantum correlations and facilitate to further explore the possibility of
secure quantum protocols in such kind of macroscopic quantum systems.
Motivated by these works, we study the quantum correlations and Gaussian
quantum steering between two squeezed magnon modes of two yttrium iron garnet
(YIG) spheres in a system of two spatially separated microwave cavities. Each
cavity also contains an optical parametric amplifier (OPA) as well as a
macroscopic YIG sphere placed near the maximum bias magnetic fields such that
this leads to the excitation of the corresponding magnon mode and its coupling
with the cavity mode. In addition, both the cavities are simultaneously driven
by a two mode squeezed vacuum microwave field in our proposed system [49]. In
this present work, we found the generation of a considerable bipartite
entanglement between the two magnon modes with gradually increasing squeezing
parameter and the mean thermal magnon number. Moreover, it can be seen clearly
from our work that not all entangled states allow for quantum steering whereas
any state that can be steered must necessarily be entangled.
This paper is organized as follows: In Section II, we introduce model
Hamiltonian and also evaluate corresponding quantum Langevin equations
including its solutions (QLEs). In Section III, we discuss in details about
mathematical formulation of bipartite entanglement and Gaussian quantum
steering between two magnon modes. Numerical Results and related discussions
are given in Section IV whereas we conclude our results in Section V.

##  II The Model Hamiltonian

Our proposed system shown in Fig. 1 consists of two microwave cavities and two
magnon modes in two YIG spheres, which are respectively placed inside the
cavities near the maximum magnetic fields of the cavity modes and
simultaneously in uniform bias magnetic fields such that it causes both the
magnon modes to strongly couple with respective cavity modes [50, 49]. Each
cavity, contains a degenerate optical parametric amplifier (OPA) to produce
squeezed light [51]. The Hamiltonian of the system in a rotating frame with
frequency $\omega_{j}$ can be written as

$\mathcal{H}=\underset{j=1,2}{\sum}\Big{\\{}\hbar\Delta_{c_{j}}c_{j}^{\dagger}c_{j}+\hbar\Delta_{m_{j}}m_{j}^{\dagger}m_{j}+\hbar
g_{j}\big{(}c_{j}m_{j}^{\dagger}+c_{j}^{\dagger}m_{j}\big{)}+ic\hbar\lambda_{j}(e^{ic\theta}c_{j}^{\dagger
2}-e^{-ic\theta}c_{j}^{2})+ic\hbar\mu_{j}(e^{ic\nu}m_{j}^{\dagger
2}-e^{-ic\nu}m_{j}^{2})\Big{\\}},$ (1)

where
$\Delta_{c_{j}}=\omega_{c_{j}}-\omega_{{}_{j}},\Delta_{m_{j}}=\omega_{m_{j}}-\omega_{{}_{j}}$,
$c_{j}$ ($c_{j}^{\dagger}$), $m_{j}$ ($m_{j}^{\dagger}$) are the annihilation
(creation) operators of the $j^{th}$ cavity and magnon modes, respectively,
and we have $\big{[}\mathcal{O},\mathcal{O}^{+}\big{]}=1$
($\mathcal{O}\,{=}\,c_{j},m_{j}$). $\omega_{c_{j}}$ ($\omega_{m_{j}}$) is the
resonance frequency of the $j^{th}$ cavity mode (magnon mode). The frequency
of the magnon mode $\omega_{m_{j}}$ is determined by the external bias
magnetic field $H_{j}$ and the gyromagnetic ratio $\beta$ via
$\omega_{m_{j}}=\beta H_{j}$, and thus can be flexibly adjusted, and $g_{j}$
is the coupling rate between the $j^{th}$ cavity and magnon modes. The
parameter $\lambda_{j}$ and $\theta_{j}$ represents respectively the nonlinear
gain of the OPA and the phase of the driving field. with $\mu_{j}$ being the
squeezing parameter and $\nu$ being the phase of $j^{th}$ squeezing mode. The
magnon squeezing can be achieved by transferring squeezing from a squeezed-
vacuum microwave field [34], or by the intrinsic nonlinearity of the
magnetostriction (the so-called ponderomotive-like squeezing) [52], or by the
anisotropy of the ferromagnet [53, 54], etc.

Figure 1: Schematic diagram of a double cavity-magnon system where both the
cavities are driven by a two-mode squeezed vacuum microwave field. Two YIG
spheres are respectively placed inside the microcavities near the maximum
magnetic fields of the cavity modes and simultaneously in uniform bias
magnetic fields.

In the frame rotating at the frequency $\omega_{{}_{j}}$, i.e., the frequency
of the $j^{th}$ mode of the input two-mode squeezed field, the QLEs of this
model Hamiltonian are given by

$\displaystyle\dot{c}_{j}$ $\displaystyle=$
$\displaystyle-(\kappa_{c_{j}}+i\Delta_{c_{j}})a_{j}-ig_{j}m_{j}+2\lambda_{j}e^{i\theta}c^{\dagger}{}_{j}+\sqrt{2\kappa_{c_{j}}}c_{j}^{in},$
(2) $\displaystyle\dot{m}_{j}$ $\displaystyle=$
$\displaystyle-(\kappa_{m_{j}}+i\Delta_{m_{j}})m_{j}-ig_{j}c_{j}+2\mu_{j}e^{ic\nu}m^{\dagger}_{j}+\sqrt{2\kappa_{m_{j}}}m_{j}^{in},$

where $\kappa_{c_{j}}$ ($\kappa_{m_{j}}$) is the decay rate of the $j$th
cavity mode (magnon mode),
$\Delta_{c_{j}}=\omega_{c_{j}}-\omega_{{}_{j}},\Delta_{m_{j}}=\omega_{m_{j}}-\omega_{{}_{j}}$,
and $c_{j}^{in}$ ($m_{j}^{in}$) is the input noise operator for the $j^{th}$
cavity mode (magnon mode). The two cavity input noise operators $c_{j}^{in}$
are quantum correlated due to the injection of the two-mode squeezed field,
and have the following non-zero correlations in time domain $c^{in}_{j}$ and
$c^{in\dagger}_{j}$ are given by [64]

$\langle
c^{in}_{j}(t)c^{in\dagger}_{j}(t^{\prime})\rangle=(\mathcal{N}+1)\delta(t-t^{\prime})$
(3) $\langle
c^{in\dagger}_{j}(t)c^{in}_{j}(t^{\prime})\rangle=\mathcal{N}\delta(t-t^{\prime})$
(4) $\langle
c^{in}_{j}(t)c^{in}_{j^{\prime}}(t^{\prime})\rangle=\mathcal{M}e^{-ic\omega_{M}(t+t^{\prime})}\delta(t-t^{\prime})\quad;\quad
j\neq j^{\prime}$ (5) $\langle
c^{in\dagger}_{j}(t)c^{in\dagger}_{j^{\prime}}(t^{\prime})\rangle=\mathcal{M}e^{ic\omega_{M}(t+t^{\prime})}\delta(t-t^{\prime})\quad;\quad
j\neq j^{\prime}$ (6)

. Here $\mathcal{N}=\sinh^{2}r$, $\mathcal{M}=\sinh r\cosh r$ and $r$ is the
squeezing parameter of the two-mode squeezed vacuum field whereas the magnon
input noise operators $m_{j}^{in}$ are of zero mean and correlated as follows

$\langle
m_{j}^{in}(t)m_{j}^{in\dagger}(t^{\prime})\rangle=(N_{m_{j}}+1)\delta(t-t^{\prime})$
(7) $\langle
m_{j}^{in\dagger}(t)m_{j}^{in}(t^{\prime})\rangle=N_{m_{j}}\delta(t-t^{\prime})$
(8)

where
$N_{m_{j}}=\left[\exp\left(\frac{\hbar\omega_{m_{j}}}{k_{B}T}\right)-1\right]^{-1}$
is the equilibrium mean thermal magnon number of the $j^{th}$ mode, with $T$
the environmental temperature and $k_{B}$ the Boltzmann constant.

Using the linearisation of quantum Langevin equations, the fluctuations of the
system are written as

$\displaystyle\delta\dot{c}_{j}$ $\displaystyle=$
$\displaystyle-(\kappa_{c_{j}}+i\Delta_{c_{j}})\delta c_{j}-ig_{j}\delta
m_{j}+2\lambda_{j}e^{i\theta}\delta{c}^{\dagger}_{j}+\sqrt{2\kappa_{c_{j}}}c_{j}^{in},$
(9) $\displaystyle\delta\dot{m}_{j}$ $\displaystyle=$
$\displaystyle-(\kappa_{m_{j}}+i\Delta_{m_{j}})\delta m_{j}-ig_{j}\delta
c_{j}+2\mu_{j}e^{ic\nu}\delta{m}^{\dagger}_{j}+\sqrt{2\kappa_{m_{j}}}m_{j}^{in}.$

To get the explicit expression of the degree of freedom of optical and magnon
modes, we consider the EPR-type quadrature fluctuations operators
corresponding to the two subsystems defined as $\delta Q_{j}=(\delta
c_{j}+\delta c_{j}^{{\dagger}})/\sqrt{2},\delta P_{j}=i(\delta
c_{j}^{{\dagger}}-\delta c_{j})/\sqrt{2},\delta q_{j}=(\delta m_{j}+\delta
m_{j}^{{\dagger}})/\sqrt{2},\delta p_{j}=i(\delta m_{j}^{{\dagger}}-\delta
m_{j})/\sqrt{2}$ (we have similar definition for input noises
$Q_{j}^{in},P_{j}^{in}$ and $q_{j}^{in},p_{j}^{in}$) [55, 56, 57, 58, 59, 60],

The above QLEs can be simplified as

$\displaystyle\delta\dot{Q}_{j}$ $\displaystyle=$
$\displaystyle-\kappa_{c_{j}}\delta Q_{j}+\Delta_{c_{j}}\delta
P_{j}+g_{j}\delta p_{j}+2\lambda\cos(\theta)\delta
q_{j}+2\lambda\sin(\theta)\delta p_{j}+\sqrt{2\kappa_{c_{j}}}Q_{j}^{in},$ (10)
$\displaystyle\delta\dot{P}_{j}$ $\displaystyle=$
$\displaystyle-\kappa_{c_{j}}\delta P_{j}-\Delta_{c_{j}}\delta
Q_{j}-g_{j}\delta q_{j}-2\lambda\cos(\theta)\delta
p_{j}+2\lambda\sin(\theta)\delta q_{j}+\sqrt{2\kappa_{c_{j}}}P_{j}^{in},$
$\displaystyle\delta\dot{q}_{j}$ $\displaystyle=$
$\displaystyle-\kappa_{m_{j}}\delta q_{j}+\Delta_{m_{j}}\delta
p_{j}+g_{j}\delta P_{j}+2\mu\cos(\nu)\delta Q_{j}+2\mu\sin(\nu)\delta
P_{j}+\sqrt{2\kappa_{m_{j}}}q_{j}^{in},$ $\displaystyle\delta\dot{p}_{j}$
$\displaystyle=$ $\displaystyle-\kappa_{m_{j}}\delta
p_{j}-\Delta_{m_{j}}\delta p_{j}-g_{j}\delta Q_{j}-2\mu\cos(\nu)\delta
P_{j}+2\mu\sin(\nu)\delta Q_{j}+\sqrt{2\kappa_{m_{j}}}p_{j}^{in}.$

Equation (10) take the following compact matrix form

$\dot{V}(t)=\mathcal{A}V(t)+\chi(t),$ (11)

Here $V(t)=[\delta Q_{1},\delta P_{1},\delta Q_{2},\delta P_{2},\delta
q_{1,}\delta p_{1},\delta q_{2},\delta p_{2}]^{T}$, $\mathcal{A}$ is the drift
matrix

$\mathcal{A}=\begin{pmatrix}\mathcal{A}_{1}&\mathcal{A}_{3}\\\
\mathcal{A}_{3}&\mathcal{A}_{2}\end{pmatrix}$ (12)

where

$\mathcal{A}_{1}=\begin{pmatrix}-\kappa_{c_{1}}+2\lambda\cos(\theta)&\Delta_{c_{1}}+2\lambda\sin(\theta)&0&0\\\
-\Delta_{c_{1}}+2\lambda\sin(\theta)&-\kappa_{c_{1}}-2\lambda\cos(\theta)&0&0\\\
0&0&-\kappa_{c_{2}}+2\lambda\cos(\theta)&\Delta_{c_{2}}+2\lambda\sin(\theta)\\\
0&0&-\Delta_{c_{2}}+2\lambda\sin(\theta)&-\kappa_{c_{2}}-2\lambda\cos(\theta)\end{pmatrix}$
(13)

and

$\mathcal{A}_{2}=\begin{pmatrix}-\kappa_{m_{1}}+2\mu\cos(\nu)&\Delta_{m_{1}}+2\mu\sin(\nu)&0&0\\\
-\Delta_{m_{1}}+2\mu\sin(\nu)&-\kappa_{m_{1}}-2\mu\cos(\nu)&0&0\\\
0&0&-\kappa_{m_{2}}+2\mu\cos(\nu)&\Delta_{m_{2}}+2\mu\sin(\nu)\\\
0&0&-\Delta_{m_{2}}+2\mu\sin(\nu)&-\kappa_{m_{2}}-2\mu\cos(\nu)\end{pmatrix}$
(14)

and

$\mathcal{A}_{3}=\begin{pmatrix}0&g_{1}&0&0\\\ -g_{1}&0&0&0\\\ 0&0&0&g_{2}\\\
0&0&-g_{2}&0\end{pmatrix}$ (15)

and
$\chi(t)=[\sqrt{2\kappa_{c_{1}}}Q_{1}^{in},\sqrt{2\kappa_{c_{1}}}P_{1}^{in},\sqrt{2\kappa_{c_{2}}}Q_{2}^{in},\sqrt{2\kappa_{c_{2}}}P_{2}^{in},\sqrt{2\kappa_{m_{1}}}q_{1}^{in},\sqrt{2\kappa_{m_{1}}}p_{1}^{in},\sqrt{2\kappa_{m_{2}}}q_{2}^{in},\sqrt{2\kappa_{m_{2}}}p_{2}^{in}]^{T}$.
The system is stable when eigenvalues of the drift matrix $\mathcal{A}$ (12)
have negative real parts. This corresponds to the so-called Routh-Hurwitz
criterion [61]. The steady state of the system, which is completely
characterized by an $8\times 8$ covariance matrix (CM) $\Sigma$, defined as
$\Sigma_{ij}(t)=\langle
V_{i}(t)V_{j}(t^{\prime})+V_{j}(t^{\prime})V_{i}(t)\rangle/2$. the solution of
$\Sigma$ can be obtained by directly solving the Lyapunov equation [62]

$\mathcal{A}\Sigma+\Sigma\mathcal{A}^{T}=-\cal{D},$ (16)

where $\mathcal{D}$ is the diffuse matrix defined by
$\mathcal{D}_{ij}\delta(t-t^{\prime})=\langle\chi_{i}(t)\chi_{j}(t^{\prime})+\chi_{j}(t^{\prime})\chi_{i}(t)\rangle/2$,
given by

$\mathcal{D}=\begin{pmatrix}\kappa^{\prime}&0&\sqrt{\kappa_{c_{1}}\kappa_{c_{2}}}\mathcal{M}&0&0&0&0&0\\\
0&\kappa^{\prime}&0&-\sqrt{\kappa_{c_{1}}\kappa_{c_{2}}}\mathcal{M}&0&0&0&0\\\
\sqrt{\kappa_{c_{1}}\kappa_{c_{2}}}\mathcal{M}&0&\kappa^{\prime\prime}&0&0&0&0&0\\\
0&-\sqrt{\kappa_{c_{1}}\kappa_{c_{2}}}\mathcal{M}&0&\kappa^{\prime\prime}&0&0&0&0\\\
0&0&0&0&\gamma^{\prime}&0&0&0\\\ 0&0&0&0&0&\gamma^{\prime}&0&0\\\
0&0&0&0&0&0&\gamma^{\prime\prime}&0\\\
0&0&0&0&0&0&0&\gamma^{\prime\prime}\end{pmatrix}$ (17)

where $\kappa^{\prime}=\kappa_{c_{1}}\big{(}\mathcal{N}+\frac{1}{2}\big{)}$,
$\kappa^{\prime\prime}=\kappa_{c_{2}}\big{(}\mathcal{N}+\frac{1}{2}\big{)}$,
$\gamma^{\prime}=\kappa_{m_{1}}\big{(}2N_{m_{1}}+1\big{)}$ and
$\gamma^{\prime\prime}=\kappa_{m_{2}}\big{(}2N_{m_{2}}+1\big{)}$.

The covariance matrix $\Sigma_{(mm)}$ associated with the two magnon modes is
given by

$\Sigma_{(mm)}=\begin{pmatrix}\mathcal{X}&\mathcal{Z}\\\
\mathcal{Z}^{T}&\mathcal{Y}\end{pmatrix}$ (18)

The $2\times 2$ sub-matrices $\mathcal{X}$ and $\mathcal{Y}$ in Eq. (18)
describe the autocorrelations of the two magnon modes and $2\times 2$ sub-
matrix $\mathcal{Z}$ in Eq. (18) denotes the cross-correlations of the two
magnon modes.

##  III Quantum correlations

###  III.1 Quantum entanglement

The logarithmic negativity $E_{m}$ is a measure or witness of entanglement in
bipartite continuous-variable (CV) systems [63]. Mathematically, it can be
expressed as:

$E_{m}=\max[0,-\log(2\psi^{-})]$ (19)

with $\psi^{-}$ being the smallest symplectic eigenvalue of partial transposed
covariance matrix $\Sigma_{(mm)}$ of two magnon modes

$\psi^{-}=\sqrt{\frac{\Gamma-\sqrt{\Gamma^{2}-4\det\Sigma_{(mm)}}}{2}}$ (20)

where the symbol $\Gamma$ is written as
$\Gamma=\det\mathcal{X}+\det\mathcal{Y}-\det\mathcal{Z}$. The two magnon modes
are entangled if the condition $\psi^{-}<1/2$ (i.e. when $E_{m}>0$) is
satisfied.

###  III.2 Gaussian quantum steering

Quantum steering is the process of acquiring information about an unmeasurable
quantum system by measuring a single quantum system. Gaussian quantum steering
is the asymmetric property between two entangled observers (the two mechanical
modes), Alice ($A$: magnon $M_{1}$) and Bob ($B$: magnon $M_{2}$). Besides,
they are used to quantify how much the two magnon modes are steerable. We use
the covariance matrix $\Sigma_{(mm)}$ of the two magnon modes, the Gaussian
steering $A\to B$ and $B\to A$ written as [Ioan, 12]

$S^{A\to B}=S^{B\to
A}=\max\left[0,\frac{1}{2}\ln{\left(\frac{\det(\mathcal{X})}{4\det\Sigma_{(mm)}}\right)}\right];$
(21)

There are two possibilities of steerability between $A$ and $B$ : (i) no-way
steering if $S^{A\to B}=S^{B\to A}=0$ i.e. Alice can’t steer Bob and vice
versa also impossible even if they are not separable, and (ii) two-way
steering if $S^{A\to B}=S^{B\to A}>0$, i.e. Alice can steer Bob and vice
versa. Indeed, a non-separable state is not always a steerable state, while a
steerable state is always not separable.

##  IV Results and Discussion

In this section, we will discuss the steady state quantum correlations of two
magnon modes under various parameters regime reported experimentally [49, 64]
and given as $\omega_{c_{1}}/2\pi=10$ GHz, $\kappa_{c}/2\pi=5$ MHz,
$\kappa_{m}=\kappa_{c}/5$, $g_{1}=g_{2}=5\kappa_{c}$, $\theta=\pi$ and
$\nu=0.9\pi$. For simplicity, we consider that $N_{m_{1}}=N_{m_{2}}=N_{m}$,
$\kappa_{c_{1}}=\kappa_{c_{2}}=\kappa_{c}$ and
$\kappa_{m_{1}}=\kappa_{m_{2}}=\kappa_{m}$. Additionally, each YIG sphere used
in our study has a diameter of 0.5 mm. These spheres are specifically chosen
for their size and contain more than $10^{17}$ spins.

Figure 2: Plot of the logarithmic negativity $E_{m}$ between two magnon modes
versus (a) $\Delta_{c_{1}}$ and $\Delta_{m_{1}}$, with
$\Delta_{c_{2}}=\Delta_{m_{2}}=0$; (b) $\Delta_{c_{2}}$ and $\Delta_{m_{2}}$,
with $\Delta_{c_{1}}=\Delta_{m_{1}}=0$, $r=1$, $\lambda=0.2\kappa_{c}$,
$\mu=0.2\kappa_{c}$ and $T=100$ mK. See text for the other parameters.

In Fig. 2, we have plotted the logarithmic negativity $E_{m}$ of subsystem
magnon-magnon with varying $\Delta_{c_{j}}$ and $\Delta_{m_{j}}$ ($j=1,2$)
whereas all other parameters are fixed. It can be seen in Figs. 2(a) and (b)
that when $\Delta_{c_{j}}=\Delta_{m_{j}}=0$ ($j=1,2$), the entanglement
between two magnon modes is optimal. This observation can be attributed to the
resonant transfer of quantum correlations from the input fields to the two
magnon modes, facilitated by the linear cavity-magnon coupling.

Figure 3: (a) Plot of the logarithmic negativity $Emm$ between the two magnon
modes vs the temperature $T(K)$ and the magnon squeezing parameter $\mu$ with
$\Delta_{c_{1,2}}=\Delta_{m_{1,2}}=0$, $T=100$ mK, $\lambda=0.2\kappa_{c}$ and
$\mu=0.2\kappa_{c}$. (b) Plot of the logarithmic negativity $Emm$ between the
two magnon modes vs $g_{2}/g_{1}$ and $r$ of the two-mode squeezed vacuum
field with $g_{1}=5\kappa_{c}$, $\Delta_{c_{1,2}}=\Delta_{m_{1,2}}=0$, $T=100$
mK, $\lambda=0.2\kappa_{c}$ and $\mu=0.2\kappa_{c}$.

Fig. 3(a) shows that the bipartite entanglement $E_{m}$ increases with the
squeezing parameter $r$ of the two-mode input squeezed vacuum field and
decreases with the temperature $T$. The effect of the temperature is due to
the influence of the decoherence phenomenon [65]. Furthermore, it has been
observed that the logarithmic negativity $E_{m}$ reaches its maximum value
when the value of $r$ falls within the range of (1-1.5) whereas when the
parameter $r$ goes to zero both the magnon modes remain separable ($E_{m}=0$)
as illustrated in figure 3(a). This shows the dependence of the bipartite
entanglement of the two magnon modes on the squeezing parameter $r$. In Fig.
3(b), we plot $E_{m}$ as a function of squeezing parameter $r$ and
$g_{2}/g_{1}$. We found here the generation of the bipartite entanglement
between the two magnon modes with a gradual increase in squeezing parameter
$r$ even for a wide range of mismatch of the two coupling strengths $g_{2}$
and $g_{1}$.

Figure 4: Plots of the steering $S^{A\to B}$, $S^{B\to A}$ logarithmic
negativity $E_{m}$ of the two magnon modes versus the equilibrium mean thermal
magnon number $N_{m}$ for various values of the coupling $\lambda$ and $\mu$,
with $\Delta_{c_{1,2}}=\Delta_{m_{1,2}}=0$, $r=1$ and $T=100$ mK.

In Fig.(4), we plot the Gaussian steering $S^{A\to B}$, $S^{B\to A}$ and
logarithmic negativity $E_{m}$ for the subsystem magnon-magnon as a function
of the equilibrium mean thermal magnon number $N_{m}$ for various values of
the parameters $\lambda$ and $\mu$ whereas the other parameters are fixed. It
can be seen that $S^{A\to B}$, $S^{B\to A}$ and entanglement $E_{m}$ have the
same evolution behavior. This figure studies the effect of $N_{m}$
(temperature $T$) and the parameters $\lambda$ and $\mu$ on the bipartite
entanglement and quantum steerings. Due to decoherence phenomena both the
quantities i.e. bipartite entanglement and quantum steering decrease very
quickly with increasing $N_{m}$. Moreover, when we enhance $\lambda$ and $\mu$
the magnon-magnon entanglement as well as two-way quantum steering become
finite for a wide range of temperature $T$ (the equilibrium mean thermal
magnon number $N_{m}$ ). Moreover, as depicted in Fig.(4) the entangled state
is not always a steerable state but a steerable state must be entangled, i.e.
when $S^{A\to B}=S^{B\to A}>0$ which leads to $E_{m}>0$ and hence is the
witnesses of the existence of Gaussian two-way steering. This means that both
magnon modes are entangled as well as are steerable from $A$ to $B$ and from
$B$ to $A$. However, we get no-way steering when $S^{A\to B}=S^{B\to A}=0$ and
$E_{m}>0$ and so the measure of Gaussian steering always remains bounded by
the bipartite entanglement $E_{m}$.

Figure 5: Plot of the logarithmic negativity $E_{m}$ between the two magnon
modes vs $\kappa_{c}$ and $\Delta_{m_{1}}$ with
$\Delta_{c_{1}}=\Delta_{c_{2}}=\Delta_{m_{2}}=0$, $T=50$ mK, $r=1.5$,
$g_{1}=5\kappa_{c}$, $\lambda=0.2\kappa_{c}$ and $\mu=0.2\kappa_{c}$.

In Fig. 5, we have plotted the logarithmic negativity $E_{m}$ of two magnon
modes with $\kappa_{c}$ and $\Delta_{m_{1}}$ for a fixed value of all other
parameters. It can be seen that the entanglement between the two magnon modes
is maximum when $\Delta_{m_{1}}=0$ and $\kappa_{c}=3\times 10^{7}$ Hz.
However, the bipartite entanglement $E_{m}$ decreases with decreasing decay
rate $\kappa_{c}$ and increasing detuning $\Delta_{m_{1}}$.

##  V Conclusions

We have theoretically investigated a scheme for the generation of the
bipartite entanglement and Gaussian quantum steering in a double microwave
cavity-magnon hybrid system where a two-mode squeezed microwave vacuum field
is also transferred simultaneously into both cavities. We have obtained
optimal parameter regimes for achieving the strong magnon-magnon entanglement
and also explored the effectiveness of the scheme towards the mismatch of two
cavity magnon couplings including the entanglement transfer efficiency. Our
present study of bipartite entangled states of two magnon modes in coupled
microwave resonators has important applications in coherent control of various
non classical correlations in macroscopic quantum systems and further
applications of such systems in quantum information processing as well as
quantum communication.

## References

  * [1] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865–942 (2009).
  * [2] R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Guhne, Rev. Mod. Phys. 92, 015001 (2020).
  * [3] C. M. Li, et al. Phys. Rev. Lett. 115, 010402 (2015).
  * [4] T. Pellizzari, S. A. Gardiner, J. I. Cirac, P. Zoller, Phys. Rev. Lett. 75, 3788 (1995).
  * [5] T. Jennewein, C.Simon, G. Weihs, H. Weinfurter, A. Zeilinger, Phys. Rev. Lett. 84, 4729 (2000).
  * [6] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993).
  * [7] J. A. Jones, D. D Jaksch,: Quantum Information. Cambridge University Press, Cambridge (2012).
  * [8] M. Neeley, et al. Nature (London) 467, 570 (2010).
  * [9] B. Julsgaard, A. Kozhekin, E.S Polzik, Nature (London) 413, 400 (2001).
  * [10] S. Mancini, D. Vitali, P. Tombesi, Phys. Rev. Lett. 90, 13 (2003).
  * [11] S. K. Singh, J. X Peng, M. Asjad, M. Mazaheri, J Phys B: At Mol Opt Phys 54, 21 (2021).
  * [12] A. Sohail, Z. Abbas, R. Ahmed, A. Shahzad, N. Akhtar, J.X. Peng, Annalen Der Physik 535, 2300087 (2023).
  * [13] B. Teklu, T. Byrnes T and F. S. Khan, Phys. Rev. A 97, 023829 (2018).
  * [14] J. Li, G. Li, S. Zippilli, D. Vitali, T. C. Zhang, Phys. Rev. A 95, 043819 (2017).
  * [15] S. K. Singh and C. H. Raymond Ooi, J. Opt. Soc. Am. B 31, 2390, (2014).
  * [16] S.K. Singh and S. V. Muniandy, Int. J. Theor. Phys. 55, 287 (2016)
  * [17] A. Hidki, Y. L. Ren, A. Lakhfif, J. El Qars, and M. Nassik, Physics Letters A 463, 128667 (2023).
  * [18] A. Hidki, Y. L. Ren, A. Lakhfif, J. El Qars, and M. Nassik, Journal of Physics B: Atomic, Molecular and Optical Physics, 56(11), 115401 (2023).
  * [19] M. Amazioug, B. Teklu, and M. Asjad, Scientific Reports 13, 3833 (2023).
  * [20] Z. B. Yang, R. C. Yang, H. Y. Liu, Quantum Inf. Process. 19, 264 (2020).
  * [21] H. Xie, L. W. He, C. G. Liao, Z. H. Chen, and X. M. Lin, Optics Express 31, 7994 (2023).
  * [22] J. Li, S. Y. Zhu, and G. S. Agarwal, Phys. Rev. Lett. 121, 203601 (2018).
  * [23] X. Zhang, C. L. Zou, L. Jiang, H. X. Tang, Phys. Rev. Lett. 113, 156401 (2014).
  * [24] D. Zhang, X. M. Wang, T. F. Li, X. Q. Luo, W. Wu, F. Nori, J. Q. You, npj Quantum. Inf. 1, 1 (2015).
  * [25] L. Bai et al., Phys. Rev. Lett. 114, 227201 (2015).
  * [26] J. Bourhill et al., Phys. Rev. B 93, 144420 (2016).
  * [27] A. Sohail, R. Ahmed, A. Shahzad, and M. A. Khan, Int J Theor Phys 61, 174 (2022).
  * [28] Y. Tabuchi et al., Science 349, 405 (2015).
  * [29] D. Lachance - Quirion et al., Sci. Adv. 3, e1603150 (2017).
  * [30] C. Kong, H. Xiong,Y. Wu, Phys. Rev. Appl. 12(3), 034001 (2019).
  * [31] Kamran Ullah, M. Tahir Naseem, and Ozgür E. Mustecaplıoglu Phys. Rev. A 102, 33721 (2020)
  * [32] Bao Wang, Zeng-Xing Liu, Cui Kong, Hao Xiong and Ying Wu, Opt. Express 26, 20248–20257 (2018).
  * [33] C. Kong, B. Wang, Z.X Liu, H. Xiong, Y. Wu, Opt. Express 27, 5544 (2019).
  * [34] A. Sohail, R. Ahmed, J. X. Peng, T. Munir, A. Shahzad, S. K. Singh, and M. C. de Oliveira, J. Appl. Phys. 133, 154401, (2023).
  * [35] J. Li, S.Y . Zhu, and G. S. Agarwal, Phys. Rev. A 99, 21801 (2019).
  * [36] M. Asjad, J. Li, S. Y. Zhu, and J. Q. You, Fundam. Res., 3, 3 (2023).
  * [37] Z. B. Yang, H. Jin, J. W. Jin, J.Y . Liu, H. Y. Liu, and R. C. Yang, Phys. Rev. Res. 3, 023126 (2021).
  * [38] A. Hidki, A. Lakhfif, J. El Qars, and M. Nassik, J. Mod. Opt. 70, 189-196 (2023).
  * [39] A. Hidki, A. Lakhfif, J. El Qars, and M. Nassik, Eur. Phys. J. D 76, 64 (2022).
  * [40] A. Hidki, A. Lakhfif, J. El Qars, and M. Nassik, Modern Physics Letter A 38, 2350044 (2023).
  * [41] X. Zhang, N. Zhu, C. L. Zou, and H. X. Tang, Phys. Rev. Lett. 117, 123605 (2016).
  * [42] X. Zhang, C. L. Zou, L. Jiang, H. X. Tang, Sci. Adv. 2, e1501286 (2016).
  * [43] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami,Y. Nakamura, Appl. Phys. Express 12, 070101 (2019).
  * [44] G. Adesso, A. Serafini, and F. Illuminati, Phys. Rev. A 70, 022318 (2004).
  * [45] J. Li, S.Y. Zhu, New J. Phys. 21, 085001 (2019).
  * [46] A. Sohail, R. Ahmed, J. X. Peng, A. Shahzad, and S. K. Singh, J. Opt. Soc. Am. B 40, 1359-1366 (2023).
  * [47] A. Sohail Shah, R. Ahmed, R. Zainab, and C. Yu, C. S., Physica Scripta. 97, 075102 (2022).
  * [48] A. Sohail, A, Hassan, R. Ahmed, et al. Quantum Inf Process 21, 207 (2022).
  * [49] Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, Y. Nakamura, Phys. Rev. Lett. 113, 083603 (2014).
  * [50] H. Huebl et al., Phys. Rev. Lett. 111, 127003 (2013).
  * [51] D. F. Walls, G. J. Milburn. Springer, Berlin (1994).
  * [52] J. Li, Y. P. Wang, J. Q. You, and S. Y. Zhu, arXiv:2101.02796. Nat. Sci. Rev. (to be published).
  * [53] H. Y. Yuan, Y. Cao, A. Kamra, R. A. Duine, and P. Yan, arXiv: 2111. 14241.
  * [54] A. Kamra and W. Belzig, Phys. Rev. Lett. 116, 146601 (2016).
  * [55] A. Sohail, R. Arif, N. Akhtar Ziauddin, J. X. Peng, G. Xianlong, and Z. D. Gu, Eur. Phys. J. Plus 138 417, (2023).
  * [56] F.N. Yusoff, M.A. Zulkifli, N. Ali, S.K. Singh, N. Abdullah, N.A.M.H. Ahmad, C.O. Edet, Photonics 10, 279 (2023)
  * [57] S. K. Singh, M. Mazaheri, J. X. Peng, A. Sohail, M. Khalid, and M. Asjad, Frontiers in Physics 11, 245 (2023).
  * [58] S. K. Singh, M. Asjad, and C. H. R. Ooi, Quantum Inf. Process. 21, 47 (2022).
  * [59] S. K. Singh, M. Mazaheri, J. X. Peng, A. Sohail, Zhidong Gu and M. Asjad, Quantum Inf. Process. 22, 198 (2023).
  * [60] S. K. Singh, M. Parvez, T. Abbas, J. X. Peng, M. Mazaheri, and M. Asjad, Phys. lett. A 442, 128181 (2022).
  * [61] E. X. DeJesus, C. Kaufman, Phys. Rev. A 35, 5288 (1987).
  * [62] D. Vitali, S. Gigan, A. Ferreira, H.R. Bohm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, M. Aspelmeyer, Phys. Rev. Lett. 98, 030405 (2007).
  * [63] G. Adesso, A. Serafini, F. Illuminati, Phys. Rev. A 70, 022318 (2004).
  * [64] M. Yu, S. Y. Zhu, and J. Li, Journal of Physics B: Atomic, Molecular and Optical Physics, 53 (6), 065402 (2020).
  * [65] W. H. Zurek, Rev. Mod. Phys. 75, 715 (2003).