# Coherent feedback control of quantum correlations in cavity magnomechanical
system with magnon squeezing

M. Amazioug  LPTHE-Department of Physics, Faculty of sciences, Ibn Zohr
University, Agadir, Morocco  S. K. Singh  Graphene and Advanced 2D Materials
Research Group (GAMRG), School of Engineering and Technology, Sunway
University, Selangor Malaysia  B. Teklu  Department of Applied Mathematics and
Sciences, Khalifa University, Abu Dhabi 127788, UAE  Center for Cyber-Physical
Systems (C2PS), Khalifa University, 127788 Abu Dhabi, UAE  M. Asjad
Department of Applied Mathematics and Sciences, Khalifa University, Abu Dhabi
127788, UAE

###### Abstract

We address a scheme to enhance the quantum correlations in cavity opto-
magnomechanical system by using the coherent feedback loop in the presence of
magnon squeezing. The proposed coherent feedback-control allows a significant
enhancement of the entanglement of three bipartite subsystems, i.e., photon-
phonon, photon-magnon and phonon-magnon. We also study the Einstein-Podolsky-
Rosen steering and one-way steering in the presence of thermal effects without
imposing additional conditions of asymmetric losses or noises in the
subsystems. Furthermore, we investigate the sensitiveness of the scheme to the
magnon squeezing, and its performance in non-ideal situations in which losses
and noises are taken into account.  Keywords : Cavity magnomechanics, Coherent
feedback, Entanglement, Steerability.

##  I Introduction

Entanglement and Einstein-Podolsky-Rosen (EPR) steering are two quantum
resources, which play a crucial role in quantum information processing and
communication. Quantum entanglement plays an important role in various
applications in quantum information processing, such as quantum teleportation
[1], superdense coding [2], telecloning [3] and quantum cryptography [4]. Many
schemes have been proposed over the past decades for processing quantum
information such as spins [5, 6], ions[7, 8, 9, 10], atoms [11, 12, 13, 14,
15, 16], photons[17, 18, 19, 20, 21, 22, 23, 24, 25], phonons [26, 27, 28].
Besides, quantum steering is a class of asymmetric quantum correlations
stronger than entanglement [29] but weaker than the violation of Bell’s
inequality [30]. The concept of quantum steering was introduced first by
Schrödinger in the context of the EPR parado [31, 32] and it can be asymmetric
(one-way), and symmetric (two-way) [33]. Steering is then a natural resource
for one sided device-independent quantum key distribution [34, 35].

In recent years, magnons, as the quanta of collective spin excitations in
yttrium iron garnet $(Y_{3}Fe_{5}O_{12},YIG)$, are of paramount importance
role due to their high spin density, low damping rate and great tunability.
Therefore, cavity magnomechanics has attracted considerable attention and
offers a robust platform where ferrimagnetic cristal (e.g., yttrium iron
garnet $(YIG)$ sphere) is coupled with a microwave cavity [36, 37]. In the
cavity magnomechanics, a magnon mode (spin wave) is combined with a vibratory
deformation mode of a ferromagnet (or ferrimagnet) by the magnetostrictive
force, and a microwave cavity mode by the interaction of magnetic dipoles. The
magnetostrictive interaction is a dispersive interaction similar to a
radiation pressure for a large ferromagnet, where the frequency of the
mechanical mode is very lower than the magnon frequency [41, 38].

In this paper, we consider coherent feedback technique [39, 40] to enhance the
entanglement and steerability in an opto-magnomechanical system consisting of
a cavity contaning $(YIG)$ sphere with the magnon self-Kerr nonlinearity as
shown in Fig. 1. We find a significant enhancement of quantum correlations via
magnon squeezing which is generated by using the magnon self-Kerr nonlinearity
[42, 43]. The magnon self-Kerr nonlinearity [[44]] can be generated via
coupling the magnon mode to a superconducting qubit [45]. We consider the
logarithmic negativity [46, 47] to quantify the quantum entanglement of three
bipartite subsystems. The steerability of the subsystem $A$ by the first
subsystem $B$ is used to quantify how much the two entangled bipartite states
are steerable. We discuss the enhancement of nonclassical correlations via
coherent feedback technique in the presence of the magnon self-Kerr
nonlinearity. We show the role of the feedback technique in the presence of
the magnon self-Kerr nonlinearity and when $\beta=\pi$ to make the
nonclassical correlations very robust to the thermal effects.

The paper is organized as follows. In Sec. II, we give the explicit expression
of the Hamiltonian and the corresponding nonlinear quantum Langevin equations
of the system. In Sec. III, we provide the linearized quantum Langevin
equations for the system. We present a method in Sec. IV to quantify
entanglement for two-mode continuous-variable (CV) and Gaussian quantum
steering. The results and discussions are given in Sec. V. Concluding remarks
are given in Sec. VI

##  II Model

We consider a cavity magnomechanics driven by single coherent laser source and
a microwave cavity with coherent feedback as depicted in Fig. 1.

Figure 1: Schematic diagram of a single-mode cavity with feedback loop and a
$(YIG)$ sphere with magnon self-Kerr nonliearity. The magnons are embodied by
a collective motion of a large number of spins in a macroscopic ferrimagnet,
and the magnon mode is directly driven by a microwave source (not shown) to
enhance the magnomechanical coupling. The cavity is also driven by an
electromagnetic field with amplitude $\Omega$. The photons and magnons of the
cavity are coupled by dipole magnetic interaction, and the magnons and phonons
are coupled by magnetostrictive interaction. A microwave field (not shown) is
implemented to improve magnon-phonon coupling. At the sphere $(YIG)$, the
magnetic field (along the x-axis) of the cavity mode, the driving magnetic
field (in the y-direction) and bias magnetic field (z-direction) are common
perpendicular. An input laser light field enters in the cavity across an
asymmetric beam splitter (BS). The output field is fully reflected on the $M$
mirror and some of the output field is sent to the cavity by the beam
splitter.

where a yttrium iron garnet (YIG) sphere with the diameter $250-\mu$
m-diameter (Ref. [41]) is placed inside the cavity. In this system, the
coupling between magnons and cavity photons is due to the magnetic dipole
interaction. The magnetostrictive interaction mediates the coupling between
magnons and phonons. The variable magnetisation induced by the magnon
excitation within the $(YIG)$ sphere causes the deformation of its geometric
structure, which forms the vibrational modes of the sphere, and vice versa
[48]. We consider the influence of radiation pressure to be insignificant
because the size of the sphere is much smaller than the microwave wavelength.
The Hamiltonian of the system is described by the form (with $\hbar=1$)

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\omega_{a}a^{{\dagger}}a+\omega_{b}b^{{\dagger}}b+\frac{\omega_{m}}{2}(x^{2}+y^{2})+\xi(b^{\dagger}b)^{2}+g_{Gb}b^{{\dagger}}bx$
(1) $\displaystyle+$ $\displaystyle
g_{Ga}(a+a^{{\dagger}})(b+b^{{\dagger}})+i\Omega(b^{{\dagger}}e^{-i\omega_{0}t}-be^{i\omega_{0}t})$
$\displaystyle+$ $\displaystyle
i\mathcal{E}(a^{{\dagger}}e^{-i\omega_{l}t}-ae^{i\omega_{l}t}),$

where $a$ ($a^{{\dagger}}$) and $b$ ($b^{{\dagger}}$)
($[O,O^{{\dagger}}]\,{=}\,1$, $O\,{=}\,a,b$) are the annihilation (creation)
operators of the cavity and magnon modes, respectively, $x$ and $y$
($[x,y]\,{=}\,i$) are the dimensionless position and momentum quadratures of
the mechanical mode, and $\omega_{a}$, $\omega_{b}$, and $\omega_{m}$ are
respectively the resonance frequency of the cavity, magnon and mechanical
modes. $\xi$ is the self-Kerr coefficient. The magnon frequency is determined
by the external bias magnetic field $H$ and the gyromagnetic ratio $\kappa$,
i.e., $\omega_{b}=\kappa H$. The single-magnon magnomechanical coupling rate
$g_{Gb}$ is small, but the magnomechanical interaction can be improved via
driving the magnon mode with a strong microwave field (directly driving the
$(YIG)$ sphere with a microwave source [49, 50]). The coupling rate $g_{Ga}$
between the magnon and microwave can be larger than the dissipation rates
$\gamma_{a}$ and $\gamma_{b}$ of the cavity and magnon modes respectively,
entering into the strong coupling regime, $g_{Ga}>\gamma_{a},\gamma_{b}$. In
the frame rotating at the drive frequency $\omega_{0}$ and applying the
rotating-wave approximation (RWA) of the system,
$g_{Ga}(a+a^{{\dagger}})(b+b^{{\dagger}})\to
g_{Ga}(ab^{{\dagger}}+a^{{\dagger}}b)$ (valid when $\omega_{a},\omega_{b}\gg
g_{Ga},\gamma_{a},\gamma_{G}$, which is easily satisfied [41]). The parameter
$\Omega=\frac{\sqrt{5}}{4}\kappa\\!\sqrt{N}B_{0}$ represents the Rabi
frequency [51] which describes the coupling strength of the drive magnetic
field (with $B_{0}$ and $\omega_{0}$ are respectively the amplitude and
frequency ) with the magnon mode, where $\kappa/2\pi=28$ GHz/T, and the total
number of spins $N=\rho V$ with $V$ the volume of the sphere and
$\rho=4.22\times 10^{27}$ m-3 the spin density of the $(YIG)$. The Rabi
frequency $\Omega$ is derived under the assumption of the low-lying
excitations, $\langle b^{{\dagger}}b\rangle\ll 2Ns$, with $s=\frac{5}{2}$ is
the spin number of the ground state Fe3+ ion in $(YIG)$. Then the full
dynamics in the presence of coherent feedback and noises is described by the
corresponding quantum Langevin equations (QLEs)

$\displaystyle\dot{a}$ $\displaystyle=$
$\displaystyle-(i\Delta_{fb}+\gamma_{fb})a-ig_{Ga}b-\psi\mathcal{E}+(2\gamma_{a})^{\frac{1}{2}}a_{fb}^{\rm
in},$ $\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle-(i\Delta_{b}+\gamma_{b})b-ig_{Ga}a-ig_{Gb}bx-2i\xi
b^{{\dagger}}bb+\Omega+\sqrt{2\gamma_{G}}b^{\rm in},$ $\displaystyle\dot{x}$
$\displaystyle=$ $\displaystyle\omega_{m}y,$ $\displaystyle\dot{y}$
$\displaystyle=$
$\displaystyle-\omega_{m}x-\gamma_{m}y-g_{Gb}b^{{\dagger}}b+\phi,$ (2)

where $\Delta_{b}=\omega_{b}-\omega_{0}$, $\gamma_{b}$ is the dissipation rate
of the magnon mode, $\gamma_{m}$ is the mechanical damping rate,
$\gamma_{fb}=\gamma_{a}(1-2\tau\cos{\beta})$ is the modified cavity decay rate
and $\Delta_{fb}=\Delta_{a}+2\gamma_{a}\tau\sin{\beta}$ is the effective
detuning with $\Delta_{a}=\omega_{a}-\omega_{0}$ are respectively the
effective cavity decay rate and the detuning with Here, the quantities $\psi$,
$\tau$ denote the transmission and reflection coefficients respectively and
$\beta$ describes the phase shift generated by the reflectivity of the output
field on the mirrors [52]. The operator $a^{in}_{fb}$ describes the effective
input noise operator in the presence of coherent feedback and corresponding
description is based on input-output theory [53]. Specifically it can be
written as $a^{in}_{fb}=\tau{e}^{{i}\beta}a^{out}+\psi a^{in}$, where $a^{in}$
is the input noise operator associatetd with microwave mode with only non-zero
correlations $\langle
a^{in}(t)a^{in^{\dagger}}(t^{\prime})\rangle=n_{a}(\omega_{a})\delta(t-t^{\prime})$
and $\langle
a^{in^{\dagger}}(t)a^{in}(t^{\prime})\rangle=(n_{a}(\omega_{a})+1)\delta(t-t^{\prime})$.
Then the corrosponding correlation functions for the effective input noise
operator $a^{in}_{fb}$ for the microwave mode can be written as [54]

$\displaystyle\langle a_{fb}^{\rm in}(t)\,a_{fb}^{\rm
in{\dagger}}(t^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\psi^{2}|1-\tau{e}^{{i}\beta}|^{2}[n_{a}(\omega_{a}){+}1]\,\delta(t{-}t^{\prime}),$
$\displaystyle\langle a_{fb}^{\rm in{\dagger}}(t)\,a_{fb}^{\rm
in}(t^{\prime})\rangle$ $\displaystyle=$
$\displaystyle\psi^{2}|1-\tau{e}^{{i}\beta}|^{2}n_{a}(\omega_{a})\,\delta(t{-}t^{\prime}).$
(3)

Morover, $b^{\rm in}$ and $\phi$ are input noise operators for the magnon and
mechanical modes, respectively, which are zero mean and characterized by the
following correlation functions [55]

$\displaystyle\langle b^{\rm in}(t)\,b^{\rm in{\dagger}}(t^{\prime})\rangle$
$\displaystyle=$ $\displaystyle[n_{b}(\omega_{b})+1]\,\delta(t{-}t^{\prime}),$
(4) $\displaystyle\langle b^{\rm in{\dagger}}(t)\,b^{\rm
in}(t^{\prime})\rangle$ $\displaystyle=$ $\displaystyle
n_{b}(\omega_{b})\,\delta(t{-}t^{\prime}),$ (5)
$\displaystyle\langle\phi(t)\phi(t^{\prime})\,{+}\,\phi(t^{\prime})\phi(t)\rangle/2\,\,$
$\displaystyle{\simeq}$
$\displaystyle\,\,\gamma_{b}[2n_{m}(\omega_{m}){+}1]\delta(t{-}t^{\prime}).$
(6)

The mechanical quality factor ${\cal Q}=\omega_{m}/\gamma_{m}\,\,{\gg}\,1$ is
large for a Markovian approximation [56], where
$n_{j}(\omega_{j}){=}\big{[}{\rm
exp}\big{(}\frac{\hbar\omega_{j}}{k_{B}T}\big{)}{-}1\big{]}^{-1}$
$(j{=}a,b,m)$ are the equilibrium mean thermal photon, magnon, and phonon
number, respectively.

Figure 2: (a) Density plot of bipartite entanglement between photon and magnon
modes$E_{ab}$, (b) magnon and phonon modes $E_{bm}$ and (c) cavity and phonon
modes $E_{am}$ as function of normalize detunings $\Delta_{a}/\omega_{m}$ and
$\tilde{\Delta}_{b}/\omega_{m}$ for $\tau=0.9$, $T=10$ mK, $\beta=\pi$ and
$\xi=\gamma_{a}$. See text for the other parameters.

##  III Linearization of quantum Langevin equations

Heisenberg-Langevin in Eq.(II) are non-linear in nature and generally cannot
be solved analytically. To solve analytical these equations, we use the
following linearization scheme. We re-write each operator as a sum of the
stationary state mean and a fluctuating quantum operator $O=O_{s}+\delta O$
($O\,{=}\,a,b,x,y$), and neglecting second order fluctuation terms when the
magnon mode is strongly driven (large amplitude $|\langle b\rangle|\gg 1$ at
the steady state), and the cavity field also has a large amplitude $|\langle
a\rangle|\gg 1$ via the cavity-magnon beamsplitter interaction. This gives the
steady-state solutions according to

$\displaystyle\langle b\rangle$ $\displaystyle=$ $\displaystyle\frac{\Omega-
ig_{Ga}\langle a\rangle}{i\tilde{\Delta}_{b}+\gamma_{b}},$ (7)
$\displaystyle\langle a\rangle$ $\displaystyle=$
$\displaystyle-\frac{ig_{Ga}\langle
b\rangle+i\psi\mathcal{E}}{i\Delta_{fb}+\gamma_{fb}}$ (8)

and for $|\tilde{\Delta}_{b}|,|\Delta_{fb}|\gg\gamma_{fb},\gamma_{b}$, one
gets

$\langle
b\rangle\simeq\frac{i\Omega\Delta_{fb}-i\psi\mathcal{E}}{g_{Ga}^{2}-\tilde{\Delta}_{b}\Delta_{fb}},$
(9)

where $\tilde{\Delta}_{b}=\Delta_{b}+g_{Gb}\langle x\rangle+2i\xi|\langle
b\rangle|^{2}$ is the effective magnon-drive detuning including the frequency
shift due to the magnomechanical interaction, and
$\tilde{G}_{Gb}=i\sqrt{2}g_{Gb}\langle b\rangle$ is the effective
magnomechanical coupling rate, where $\langle
x\rangle=-\frac{g_{Gb}}{\omega_{m}}\langle b\rangle^{2}$. The linearized QLEs
describing the quadrature fluctuations $\delta X_{a}=(\delta a+\delta
a^{{\dagger}})/\sqrt{2},\quad\delta Y_{a}=i(\delta a^{{\dagger}}-\delta
a)/\sqrt{2},\delta X_{b}=(\delta b+\delta b^{{\dagger}})/\sqrt{2},\delta
Y_{b}=i(\delta b^{{\dagger}}-\delta b)/\sqrt{2},\delta x$ and $\delta y$ can
be written in compact matrix form as

$\dot{u}(t)=\mathcal{L}u(t)+\mu(t),$ (10)

where $u(t)=\big{[}\delta X_{a}(t),\delta Y_{a}(t),\delta X_{b}(t),\delta
Y_{b}(t),\delta x(t),\delta y(t)\big{]}^{T}$ is vector of quadrature
fluctuation operators, $\mu(t)=\big{[}\\!\sqrt{2\gamma_{a}}X_{a}^{\rm
in}(t),\sqrt{2\gamma_{a}}Y_{a}^{\rm in}(t),\sqrt{2\gamma_{b}}X_{b}^{\rm
in}(t),\sqrt{2\gamma_{b}}Y_{b}^{\rm in}(t),0,\phi(t)\big{]}^{T}$ is the vector
of input noise operators, and the drift matrix $\mathcal{L}$ can be written as

$\mathcal{L}=\begin{pmatrix}-\gamma_{fb}&\Delta_{fb}&0&g_{Ga}&0&0\\\
-\Delta_{fb}&-\gamma_{fb}&-g_{Ga}&0&0&0\\\
0&g_{Ga}&-\gamma_{b}+\xi&\tilde{\Delta}_{b}&-\tilde{G}_{Gb}&0\\\
-g_{Ga}&0&-\tilde{\Delta}_{b}&-\gamma_{b}-\xi&0&0\\\ 0&0&0&0&0&\omega_{m}\\\
0&0&0&\tilde{G}_{Gb}&-\omega_{m}&-\gamma_{m}\\\ \end{pmatrix}.$ (11)

The drift matrix in Eq. (11) is provided under the condition
$|\tilde{\Delta}_{b}|,|\Delta_{fb}|\gg\gamma_{fb},\gamma_{b}$. In fact, we
will show later that
$|\tilde{\Delta}_{b}|,|\Delta_{fb}|\simeq\omega_{m}\gg\gamma_{fb},\gamma_{b}$
[see Fig. 1 (b)] are optimal for the presence of all bipartite entanglements
of the system. Note that Eq. (7) is intrinsically nonlinear since
$\tilde{\Delta}_{b}$ contains $|\langle b\rangle|^{2}$. However, for a given
value of $\tilde{\Delta}_{b}$ (one can always tune $\tilde{\Delta}_{b}$ by
adjusting the bias magnetic field) $\langle b\rangle$, and hence
$\tilde{G}_{Gb}$, can be achieved straightforwardly.

Figure 3: (a) Density plot of bipartite entanglement between photon and magnon
modes $E_{ab}$, (b) magnon and phonon modes $E_{bm}$ and (c) cavity and phonon
modes $E_{am}$ versus the reflectivity parameter $\tau$ and phase $\beta$ for
$\tilde{\Delta}_{b}=0.9\omega_{m}$, $\Delta_{a}=-\omega_{m}$ and
$\xi=\gamma_{a}$. See text for the other parameters.

##  IV Entanglement and steerabilities

Figure 4: (a) Plot of photon and magnon modes ($E_{ab}$), cavity and phonon
modes ($E_{am}$) and magnon and phonon modes ($E_{bm}$) as a function of
temperature $T$, self-Kerr coefficient $\xi$ (see the Fig. (b)) and
reflectivity parameter $\tau$ (see the Fig. (c)). We take $G_{Gb}/2\pi=4.8$
MHz, $\Delta_{a}=-\omega_{m}$ and $\tilde{\Delta}_{b}=0.9\omega_{m}$. The
reflectivity parameter is $\tau=0.98$ and $\tau=0.4$ in Fig. (a) and Fig. (b)
respectively. In Figs. (b)-(c) the temperature is $T=10$ mK and in Figs.
(a)-(c) the magnon self-Kerr nonlinearity coefficient is $\xi=\gamma_{a}$. See
text for the details of the other parameters.

The steady state evolution of the quantum fluctuations of the system is a
continuous variable (CV) three-mode Gaussian state is completely characterized
by a $6\times 6$ covariance matrix (CM) $\mathcal{V}$, where
$\mathcal{V}_{ij}=\frac{1}{2}\langle
u_{i}(t)u_{j}(t^{\prime})+u_{j}(t^{\prime})u_{i}(t)\rangle$ ($i,j=1,2,...,6$)
of the covariance matrix (CM) $\Delta$ satisfies [58, 57]

$\mathcal{L}\mathcal{V}(t)+\mathcal{V}(t)\mathcal{L}^{T}=-\mathcal{K},$ (12)

where $\mathcal{K}={\rm diag}\big{[}\gamma_{a}\psi^{2}|1-\tau
e^{i\beta}|^{2}(2n_{a}+1),\gamma_{a}\psi^{2}|1-\tau
e^{i\beta}|^{2}(2n_{a}+1),\gamma_{b}(2n_{b}+1),\gamma_{b}(2n_{b}+1),0,\gamma_{m}(2n_{m}+1)\big{]}$
is the diffusion matrix, which is defined through
$\langle\mu_{i}(t)\mu_{j}(t^{\prime})+\mu_{j}(t^{\prime})\mu_{i}(t)\rangle/2=\mathcal{K}_{ij}\delta(t-t^{\prime})$.
The covariance matrix $\sigma_{AB}$ of two modes $A$ and $B$ may be written as

$\sigma_{AB}=\begin{pmatrix}\mathcal{A}&\mathcal{C}\\\
\mathcal{C}^{T}&\mathcal{B}\end{pmatrix}.$ (13)

The $2\times 2$ sub-matrices $\mathcal{A}$ and $\mathcal{B}$ in Eq. (13)
describe the autocorrelations of the two modes and $2\times 2$ sub-matrix
$\mathcal{C}$ in Eq. (13) denotes the cross-correlations of the two modes.
Characterizing, quantifying and classifying quantum correlations in
multipartite quantum systems are one of the most problematic issues in quantum
information and especially in optomagnomechanical systems when information is
encoded in continuous variables (CV). The entanglement in CV system can be
quantified by by using the logrithmic negativity $E_{N}$ [46, 47]

$E_{N}=\max[0,-\log(2\Lambda^{-})],$ (14)

where
$\Lambda^{-}=\sqrt{\mathcal{X}-(\mathcal{X}^{2}-4\det\sigma)^{1/2}}/\sqrt{2}$
being the minimum symplectic eigenvalue of partially transposed covariance
matrix of two mode Gaussian states, with
$\mathcal{X}=\det\mathcal{A}+\det\mathcal{B}-\det\mathcal{C}$. The two
subsystems are entangled if $E_{N}>0$. For the two-mode Gaussian state (13),
the system is separable if $\Lambda^{-}<1/2$, where $\Lambda^{-}$ being the
smallest symplectic eigenvalue of partial transposed covariance matrix Eq.
(13). Another quantum correlation quantifier which is of paramount importance
in optomagnomechanical systems is the quantum steering. The steerability of
Bob $(B)$ by Alice $(A)$ $(A\to B)$ for a $(n_{A}+n_{B})$ mode Gaussian state
can be quantified by [33]

$S^{A\to
B}(\sigma_{AB})=\max\left[0,-\sum_{j:\bar{\nu}_{j}^{AB/A}<1}ln\left(\bar{\nu}_{j}^{AB/A}\right)\right],$
(15)

where $\bar{\nu}_{j}^{AB/A}$ $(j=1,...,m_{B})$ are the symplectic eigenvalues
of $\bar{\sigma}_{AB/A}=B-C^{T}A^{-1}C$, derived from the Schur complement of
$A$ in the covariance matrix $\sigma_{AB}$. The steerability of Alice by Bob
$[S^{A\to B}(\sigma_{AB})]$ can be obtained by swapping the roles of $A$ and
$B$. We notice that a non-separable state is not always steerable but a
steerable state is always non separable state. Thus we have two possibilities
between $A$ and $B$ : ($i$) if $\mathcal{S}^{A\to B}=\mathcal{S}^{B\to A}=0$
Alice can’t steer Bob and vice versa even if they are entangled (i.e. no-way
steering), ($ii$) if $\mathcal{S}^{A\to B}>0$ and $\mathcal{S}^{B\to A}=0$ or
$\mathcal{S}^{A\to B}=0$ and $\mathcal{S}^{B\to A}>0$ as one-way steering,
i.e. Alice can steer Bob but Bob can’t steer Alice and vice versa, and ($iii$)
if $\mathcal{S}^{A\to B}=\mathcal{S}^{B\to A}>0$ Alice can steer Bob and vice
versa (i.e. two-way steering). In addition, the measurement of Gaussian
Steering is always bounded by the entanglement. In order to check the
asymmetric steerability of the two mode Gaussian state, we introduce the
steering asymmetry which is defined as

$S(AB)=|S^{A\to B}-S^{B\to A}|.$ (16)

##  V Results and Discusion

In this section, we show the results and discuss the evolution of quantum
correlations of the system by considering experimentally accessible parameters
reported in [54, 51]: $\omega_{a}/2\pi=10$ GHz, $\omega_{m}/2\pi=10$ MHz ,
$\gamma_{m}/2\pi=100$ Hz, $\gamma_{a}/2\pi=\gamma_{b}/2\pi=1$ MHz,
$g_{Ga}/2\pi=G_{Gb}/2\pi=3.2$ MHz, and at low temperature $T=10$ mK.
$G_{Gb}=2\pi\times 3.2$ MHz implies the drive magnetic field $B_{0}\approx
3.9\times 10^{-5}$ T for $g_{Ga}\approx 2\pi\times 0.2$ Hz, corresponding to
the drive power $P=8.9$ mW.

Figure 5: Plot of bipartite entanglement, Gaussian quantum steering and
asymmetric quantum steering between photon and magnon modes (a) $E_{ab}$,
$S^{a\to b}$ and $S^{b\to a}$ and $S(ab)$, cavity and phonon modes (b)
$E_{am}$, $S^{a\to m}$ and $S^{m\to a}$ and $S(am)$ and magnon and phonon
modes (c) $E_{bm}$, $S^{b\to m}$ and $S^{m\to b}$ and $S(bm)$, Gaussian
quantum steering between photon and magnon modes ($S^{a\to b}$ and $S^{b\to
a}$) as a function of the temperature $T$ . We take $G_{Gb}/2\pi=4.8$ MHz,
$\beta=\pi$, $\tau=0.98$, $\xi=\gamma_{a}$, $\tilde{\Delta}_{b}=0.9\omega_{m}$
and $\Delta_{a}=-\omega_{m}$. See text for the details of the other
parameters.

We present in Fig. (2), the steady state of the three bipartite entanglement
$E_{ab}$ (between the cavity and magnon mode), $E_{bm}$ (between the magnon
and mechanical mode) and $E_{am}$ (between the cavity and mechanical mode)
versus the detunings $\Delta_{a}$ and $\tilde{\Delta}_{b}$ in the presence of
coherent feedback loop with the magnon self-Kerr nonlinearity. We observe,
that the entanglement is very strong ($E_{ab}>1.3$, $E_{bm}>0.8$ and
$E_{am}>1.3$) in comprising with the results in Ref. [54, 51]. The maximum
value of entanglement of the three bipartite is improves via coherent feedback
loop and the magnon self-Kerr nonlinearity when $\beta=\pi$. We remark, when
$\Delta_{a}=-\omega_{m}$ and $\tilde{\Delta}_{b}=0.9\omega_{m}$ the
entanglement $E_{ab}$ and $E_{am}$ are maximum while $E_{bm}\approx 0.2$. In
Fig. (3) we plot there bipartite entanglements $E_{ab}$, $E_{bm}$ and $E_{am}$
as a function of the reflectivity $\tau$ and $\beta$ as shown We remark that
the entanglement is increasing with $\tau$ and $\beta$ and it robust when
$\beta=\pi$. Moreover, the entanglement is achieved its maximum value when
$\gamma_{fb}=\gamma_{a}(1+2\tau)$.

In Fig. (4) we plot the entanglements of the three bipartite $E_{ab}$,
$E_{am}$ and $E_{bm}$ versus different parameters. We remark that the existing
of genuig tripartite entanglement when all bipartite entanglement are non-
vanishing as illustrated in Fig. (4). We notice, the entanglement is robust
against temperature as depicted in Fig. (4)(a) and survive above 3K. We
observe that the entanglement of all the subsystem is diminishes due to
decoherence phenomenon [60]. Moreover, the entanglement between photon-magnon
and photon-phonon persists for temperature $T>3$ K and $T\approx 2.5$ K
respectively, whereas, the entanglement between magnon-phonon is vanishes at
lower temperatures ($T\approx 0.2$ K) even this temperature is the maximum
achieved in the Ref. [51]. One can say that the entanglement between photon-
magnon and photon-phonon is stronger than the entanglement between magnon-
phonon. The entanglement between photon-magnon and magnon-phonon increases
with increasing the magnon self-Kerr nonlinearity coefficient $\xi$, instead
the entanglement between photon-phonon decreases as illustrated in Fig.
(4)(b). The entanglement $E_{ab}\approx 0.25$ for $\xi=10^{7}$ Hz in
comprising $E_{ab}\approx 0.125$ in comprising with the results in Ref. [51].
We remark in Fig. (4)(c) the enhancement of all three bipartite entanglement
by coherent feedback technique. The maximum value reached by entanglement
between photon-magnon and photon-phonon is very important than which obtained
in Ref. [54, 51].

In Fig.(5), we plot for each bipartite the entanglement, the Gaussian steering
$S^{A\to B}$, $S^{B\to A}$ and the asymmetric steering versus the temperature
$T$. The entanglement and steerabilities diminish quickly with temperature due
to the decoherence phenomenon. We note, the one way quantum steering is more
robust than two way quantum steering and it survive for a larger value of
temperature $T$. The entangled state is not always steerable state instead
steerable state must be entangled i.e. when $S^{A\to B}=S^{B\to A}>0$ and
$E_{N}>0$ is the witnesses of existence of Gaussian two-way steering, such
that the subsystem of two subsystem are entangled but are steerable only from
$A$ to $B$ and from $B$ to $A$ [33] and no-way steering appears when $S^{A\to
B}=S^{B\to A}=0$ and $E_{N}>0$ as depicted in Fig.(5)(c). The measurement of
Gaussian steering is always bounded by the entanglement $E_{N}$ as also
discussed in [61]. Finally, the asymmetric steering $SA$ is always less than
$ln(2)$, which is maximal when the state is nonsteerable in one-way i.e.
$S^{A\to B}>0$ and $S^{B\to A}=0$ or $S^{A\to B}=0$ and $S^{B\to A}>0$ and it
decreases with increasing steerability in either way [33]. In Fig.(5)(a) the
steering from the photon mode to the magnon mode $S^{a\to b}$ has a similar
behavior to $E_{N}$ it decreases from its maximum value to zero when $T>3$ K.
Besides, one-way steering appears when $T>0.2$ K, i.e. $S^{a\to b}>0$ and
$S^{b\to a}=0$ as expected in Fig. (5)(a). Moreover, the steering from the
magnon mode to the photon mode $S^{b\to a}$ is diminishes quickly to remains
zero for $T>0.2$ K as depicted in Fig.(5)(a). Otherwise, when the temperature
$T<0.2$ K, the two-way steering occurs between optical mode and the magnon
mode, i.e. $S^{a\to b}>0$ and $S^{b\to a}>0$ ($S(ab)=0$). The steerability
between photon mode and the phonon mode is always remains one-way steering,
i.e. $S^{a\to m}>0$ ($S^{m\to a}=0$) when $T>0.2$ K as implemented in
Fig.(5)(b). The steerability between the magnon mode and phonon mode is
approximately remains two-way steering and $S^{b\to m}>S^{m\to b}$ when
$T<0.10$ K and no-way steering ($S^{b\to m}=0$ and $S^{m\to b}=0$ ($S(bm)=0$)
when $T>0.10$ K as shown in Fig.(5)(c).

##  VI Conclusions

In conclusion we have studied how coherent feedback loop improves the quantum
correlations between three bipartite subsystems in the presence of the magnon
self-Kerr nonlinearity in cavity magnomechanics systems. We quantify
steerability by using Gaussian quantum steering and show that Gaussian
steering remains bounded by entanglement, i.e. the steerable modes are
strictly entangled but the entangled modes are not necessarily steerable. We
have found one way-steering between photon-magnon and photon-phonon. However,
the steerability bewteen magnon-photon is always two-way. The entanglement and
steerabilities are shown robust against the temperature, where entanglement
may persist above 3 K in the case of photon-magnon, and approximately equal
2.5 K for photon-phonon. Moreover, The entanglement and steerabilities between
magnon-phonon is fragile under thermal effects. Our proposed scheme to improve
entanglement can be of interest for various applications in quantum
information processing.

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