# Emergence and enhancement of feedback control induced quantum entanglement

M. Amazioug  LPTHE, Department of Physics, Faculty of Sciences, Ibnou Zohr
University, Agadir, Morocco  D. Dutykh  Mathematics Department, Khalifa
University of Science and Technology, 127788, Abu Dhabi, United Arab Emirates
Causal Dynamics Pty Ltd, Perth, Australia  M. Asjad<EMAIL_ADDRESS>Mathematics Department, Khalifa University of Science and Technology, 127788,
Abu Dhabi, United Arab Emirates

###### Abstract

We present a scheme for controlling quantum correlations by applying feedback
to the cavity mode that exits a cavity while interacting with a mechanical
oscillator and magnons. In a hybrid cavity magnomechanical system with a
movable mirror, the proposed coherent feedback scheme allows for the
enhancement of both bipartite and tripartite quantum correlations. Moreover,
we demonstrate that the resulting entanglement remains robust with respect to
ambient temperatures in the presence of coherent feedback control.

##  I Introduction

The phenomenon of entanglement is a fundamental characteristic of quantum
mechanics and it is used in many quantum technology applications such as
quantum teleportation [1], superdense coding [2], telecloning [3] and quantum
cryptography [4]. In this regard, the field of cavity optomechanics has
recently gained a lot of attention as a versatile platform for variety of
quantum information processing applications, particularly realizing
entanglement between mechanical and optical dgrees of freedom [5, 6], ground
state cooling of the nanomechanical resonator [7, 8], generation of two-mode
squeezing between microwave and optical fileds [9, 10, 11] and mechanical
degrees of freedom [12, 13]. In quantum meteorology, various achievements have
been made, including enhanced precision measurements [14, 15] and
gravitational waves detection [16, 17, 18].

Recently, the magnons, as the quanta of collective spin excitations in yttrium
iron garnet ($\rm{Y_{3}Fe_{5}O_{12}}$; YIG), are of paramount importance due
to their high spin density, low damping rate and great tunability. A magnon,
first introduced by Bloch [19] is the collective spin wave excitation carrying
quantized energy in a magnetically ordered ground state. Moreover, cavity
magnomechanics has attracted considerable attention and provided a robust
platform where a ferrimagnetic crystal (_i.e._ YIG sphere) is coupled with a
microwave cavity [20, 21], phonons [22, 23], and photons [24, 25, 26]. In
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 dispersion interaction similar to a
radiation pressure for a large ferromagnet, where the frequency of the
mechanical mode is very lower than the magnon frequency [27, 28]. The
macroscopic quantum entanglement has been extensively studied by exploiting
magnetostrictive interaction in cavity magnomechanical systems [22]. However,
entanglement, as a fragile quantum resource, is prone to being destroyed by
the surrounding environment. In addition, the generation of entanglement is
often limited by the amplification effect in the unstable regime. Thus, it is
crucial to design effective schemes for generating and enhancing entanglement.

Recently, coherent feedback has attracting considerable attention in quantum
information processing and quantum optics. Coherent feedback is first
discussed in all-optical cases [29, 30]. In optomechanics consisting of a
Fabry-Pérot cavity, coherent feedback is dependent on a light field input that
travels through an asymmetric beam splitter to reach the cavity. A part of the
output electromagnetic field is partially reflected on the mirror and then
feed it back to the cavity via a asymmetric beam splitter. Feedback controle
is at the heart of a large field of applications such as to cool a mechanical
oscillator [31, 32, 33] and entanglement superconducting qubits[34].
Implementation of coherent feedback is currently realized in optomechanical
systems [35, 36, 37, 38], trapped ion [39], quantum state transfer [40] and
entanglement [41]. Addationaly, parametric feedback is implemented for cooling
and trapping single atoms [42] and trapped nanospheres [43, 44].

In this paper presents a scheme to control quantum correlations by deploying
feedback on the cavity mode that exits the cavity interacting with magnons and
a mechanical oscillator. We investigate that both bipartite and tripartite
entanglement can be enhanced by implementing a coherent feedback control in an
hybrid cavity magnomechanical system, measured using the logarithmic
negativity [45, 46, 47, 48]. Specifically, we show that the resulting
entanglement can be controlled and improved even in the presence of thermal
noises by adjusting the parameters associated with the feedback loop.

The paper is organized as follows. In Section II, we introduce the model of
the hybrid cavity-magnomechanical system under consideration and calculate the
quantum Langevin equations along with the covariance matrix for the system. In
Section III we discuss the stationary entanglement of the mechanical resonator
with the mechanical mode of the YIG sphere ($\mathcal{E}_{\rm db}$), the
magnon mode ($\mathcal{E}_{\rm dm}$), and the cavity mode ($\mathcal{E}_{\rm
da}$) as functions of various parameters of the system. In Section III we
study genuine tripartite entanglement. We further study the effects of
mechanical thermal noise and parameters associated with feed back on the
observation of the tripartite quantum correlation. Section IV contains the
main concluding remarks.

Figure 1: (a) Schematics of cavity magnomechanics with movable end mirror via
coherent feedback loop. A microwave field (not shown) is implemented to
improve magnon-phonon coupling. At YIG sphere site, the magnetic field (along
_x_ -axis) of the cavity mode, the drive magnetic field (in _y_ -direction),
and biased magnetic field (_z_ -direction) are common perpendicular. In
addition, the coherent feedback loop is achieved through the beam-splitter,
where the quantity $t$ and $\tau$ denote the real amplitude transmission and
reflection parameters. An input laser field enters in the cavity through an
asymmetric beam splitter (BS). The output field is totally reflected on the
mirror $M$ and a part of the output field is sent to the cavity via beam
splitter.

##  II The model

We consider a cavity-magnomechanical (CMM) system consisting of a single
ferrimagnetic YIG sphere and a microwave cavity, where the YIG sphere and the
microwave cavity are driven separately, as shown in Fig. 1. The microwave
cavity is composed by a fixed mirror (spatially transmitting) and the movable
mirror (perfectly reflecting). Due to the bias of the magnetic field along the
_y_ -axis, YIG sphere excites a large number of magnon modes. At the same
time, the magnetization of the YIG sphere causes its deformation, thereby
stimulating the mechanical mode of vibration. The magnons are the quanta of a
spin wave in a magnetically ordered state. A magnon mode can be the uniform-
precession Kittel mode in the YIG sphere [28, 49]. The magnon mode
simultaneously couples to a microwave cavity mode via the magnetic-dipole
interaction and to a deformation vibration mode via the magnetostrictive-
interaction (a radiation pressure-effect-liked). Note that since the
wavelength of microwaves is generally much larger than the size of YIG
spheres, the radiation pressure coupling of microwave fields to YIG sphere can
be safely ignored. The Hamiltonian of the CMM system in a rotating frame at
the magnon driving frequency $\omega_{0}$ reads as ($\hbar=1$ )

$\displaystyle\mathcal{H}$
$\displaystyle=\Delta_{a}a^{\dagger}a+\Delta_{m}m^{\dagger}m+\omega_{d}d^{\dagger}d+\omega_{b}b^{\dagger}b+g(a^{\dagger}m+am^{\dagger})+$
$\displaystyle
g_{a}a^{\dagger}a(d+d^{\dagger})+g_{m}m^{\dagger}m(b+b^{\dagger})+{\rm
i}\left(Em^{\dagger}+\mathcal{E}ta^{\dagger}-H.c\right),$

where $\Delta_{a}:=\omega_{a}-\omega_{0}$ and
$\Delta_{m}:=\omega_{m}-\omega_{0}$ are the cavity and magnon detunings. The
operators $a$, $m$, $b$, and $d$ ($a^{\dagger}$, $m^{\dagger}$, $b^{\dagger}$,
and $d^{\dagger}$) are the annihilation (creation) operators related to the
cavity, magnon, mechanical mode of the magnon, and movable end mirror. Here,
$g_{\rm a}$ is the coupling of the microwave cavity mode with the movable
mirror, $g$ is the coupling of the cavity with the magnonic mode, and $g_{\rm
m}$ is the bare magnomechanical coupling which can be significantly improved
by directly driving the magnon mode with a microwave field with the coupling
strength $E$. The cavity deriving $\mathcal{E}$ is related to the input power
$P_{L}$ by $\mathcal{E}=\sqrt{\frac{2\gamma_{a}P_{L}}{\hbar\omega_{0}}}$,
represents the coupling strength between the cavity and the driving field,
with $P_{L}(\omega_{0})$ being the power (frequency) of the input laser filed,
and $\gamma_{a}$ the cavity decay rate. Here $t$ and $\tau$ represent the real
amplitude transmission parameters of the beam splitter, and they satisfy the
relation $t^{2}+\tau^{2}=1$ (with both $t$ and $\tau$ being real and
positive).

In order to describe the complete dynamics of the system we include the
dissipation effects. Additionally to the Hamiltonian’s description of the
system in Eq.(II), it is subject to noise forces that result from quantum
fluctuations of the radiation field and fluctuations of the mechanical and
magnon mode’s thermal bath. In order to describe the complete dynamics of
subsystems involved in this problem, it is an appropriate choice to use the
quantum Langevin equations. Therefore, Heisenberg-Langevin equations of motion
for optical, mechanical, microwave, and magnon modes are given, respectively,
by

$\displaystyle\dot{a}$ $\displaystyle=$ $\displaystyle-(\gamma_{\rm fb}+{\rm
i}\Delta_{\rm fb})a-{\rm i}gm-{\rm i}g_{\rm
a}a(d+d^{\dagger})+\\!\\!t\mathcal{E}e^{{\rm i}\phi}+\sqrt{2\gamma_{\rm
a}}a_{\rm fb}^{\rm in},$ $\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle-(\gamma_{\rm b}+{\rm i}\omega_{\rm b})b-{\rm i}g_{\rm
m}m^{\dagger}m+\sqrt{2\gamma_{b}}b_{\rm in},$ $\displaystyle\dot{m}$
$\displaystyle=$ $\displaystyle-(\gamma_{\rm m}+{\rm i}\Delta_{\rm m})m-{\rm
i}ga-{\rm i}g_{\rm m}(b+b^{\dagger})m+E+\sqrt{2\gamma_{\rm m}}m_{\rm in},$
$\displaystyle\dot{d}$ $\displaystyle=$ $\displaystyle-(\gamma_{\rm d}+{\rm
i}\omega_{\rm d})d-{\rm i}g_{\rm a}a^{\dagger}a+\sqrt{2\gamma_{\rm d}}d_{\rm
in},$ (2)

where $\gamma_{\rm\cal{O}}$ (${\cal{O}}=\\{d,a,b,m\\}$) is the damping rate of
the corresponding mode, and ${\cal{O}}_{\rm in}(t)$ denotes the input noise,
which is zero-mean $\langle{\rm\cal{O}}_{\rm in}(t)\rangle=0$. For a large
value of the mechanical quality factor $Q_{\rm b}=\omega_{\rm b}/\gamma_{\rm
b}\gg 1$ and $Q_{\rm d}=\omega_{\rm d}/\gamma_{\rm d}\gg 1$ [50], the
mechanical baths can be assumed of Markovian type. Here, $\gamma_{\rm
fb}=\gamma_{\rm a}(1-2\tau\cos{\phi})$ and $\Delta_{\rm fb}=\Delta_{\rm
a}-2\gamma_{\rm a}\tau\sin{\phi}$ are, respectively, the effective cavity
decay rate and the detuning with $\phi$ describes the phase shift generated by
the reflectivity of the output field on the mirrors. The output field $a^{\rm
out}$ and the cavity field $a$ are related via standard input-output relation
$a^{\rm out}=\sqrt{2\gamma_{\rm a}}a-ta^{\rm in}$ [51]. Then, the effective
input noise operator $a^{\rm in}_{\rm fb}$, describing the input field induced
via the coherent feedback, is given by $a^{\rm in}_{\rm fb}=\tau e^{{\rm
i}\phi}a^{\rm out}+ta^{\rm in}$ which is characterized by the non-zero
correlations

$\displaystyle\langle a_{\rm fb}^{\rm in}(t)\,a_{\rm fb}^{\rm
in{\dagger}}(t^{\prime})\rangle$ $\displaystyle=$ $\displaystyle t^{2}|1-\tau
e^{{\rm i}\phi}|^{2}[N_{\rm a}(\gamma_{\rm a}){+}1]\,\delta(t{-}t^{\prime}),$
(3) $\displaystyle\langle a_{\rm fb}^{\rm in{\dagger}}(t)\,a_{\rm fb}^{\rm
in}(t^{\prime})\rangle\,\,$ $\displaystyle{=}$ $\displaystyle\,\,t^{2}|1-\tau
e^{{\rm i}\phi}|^{2}N_{\rm a}(\gamma_{\rm a})\,\delta(t{-}t^{\prime}),$ (4)

where $N_{\rm a}=\big{(}{\rm e}^{\hbar\omega_{a}/k_{B}T}-1\big{)}^{-1}$. Here,
${\rm\cal{O}}_{\rm in}(t)$ (${\rm\cal{O}}\in\\{d,b,m\\}$) is the input nose
operator with only nonzero correlations $\langle{\rm\cal{O}}_{\rm
in}(t){\rm\cal{O}}^{\dagger}_{\rm
in}(t^{\prime})\rangle=(N_{{\rm\cal{O}}}+1)\delta(t-t^{\prime})$ and
$\langle{\rm\cal{O}}^{\dagger}_{\rm in}(t){\rm\cal{O}}_{\rm
in}(t^{\prime})\rangle=N_{{\rm\cal{O}}}\delta(t-t^{\prime})$, where
$N_{{\rm\cal{O}}}=\big{(}{\rm
e}^{\hbar\omega_{k}/k_{B}T_{\rm\cal{O}}}-1\big{)}^{-1}$ being the mean thermal
occupation number with $k_{B}$ is the Boltzmann constant and $T_{\rm\cal{O}}$
is the ambient temperature for ${\rm\cal{O}}^{\rm th}$ mode.

Figure 2: (a) Density plot of the Logarithmic negativity $\mathcal{E}_{\rm
db}$, (b) $\mathcal{E}_{\rm dm}$ and (c) $\mathcal{E}_{\rm da}$ as a function
of reflection coefficient $\tau$ and phase $\phi/\pi$ for a fixed value of
detuning $\Delta_{\rm m}=\Delta_{\rm a}=\omega_{\rm b}$.The temperature of the
two phononic environments is fixed at $T_{\rm d}=T_{\rm b}=100$ mK.

The nonlinear quantum Langevin equations (QLEs) are in general non-solvable
analytically. Using the strong pump for the magnon mode, the QLEs (2) can be
linearized by writing each operator $\mathcal{O}$
(${\rm\mathcal{O}}\in\\{d,a,b,m\\}$) as the sum of its steady-state value and
the corresponding small fluctuation operators with zero mean value, _i.e._
$\mathcal{O}\equiv\delta\mathcal{O}+\ \mathcal{O}_{s}$. The QLEs (2) are then
separated into two sets of equations for classical averages and for quantum
fluctuations, respectively. The steady-state averages can be obtained by
solving the following equations

$\displaystyle a_{\rm s}$ $\displaystyle=$ $\displaystyle-{\rm i}(g\,m_{\rm
s}+t\mathcal{E}e^{{\rm i}\phi})/(\gamma_{\rm fb}+{\rm i}\tilde{\Delta}_{\rm
fb}),b_{\rm s}=-{\rm i}g_{\rm m}|m_{\rm s}|^{2}/(\gamma_{\rm b}+{\rm
i}\omega_{\rm b}),$ $\displaystyle d_{\rm s}$ $\displaystyle=$
$\displaystyle-{\rm i}g_{\rm a}|a_{\rm s}|^{2}/(\gamma_{\rm d}+{\rm
i}\omega_{\rm d}),\quad m_{\rm s}={(E-{\rm i}ga_{\rm s})/(\gamma_{\rm m}+{\rm
i}\tilde{\Delta}_{\rm m})},$ (5)

where $\tilde{\Delta}_{\rm fb}=\Delta_{\rm fb}+2g_{a}{\rm Re}[d_{\rm s}]$ and
$\tilde{\Delta}_{\rm m}=\Delta_{\rm m}+2g_{\rm m}{\rm Re}\,[b_{\rm s}]$ is the
effective detuning by including the magnetostriction. Typically, these
frequency shifts are small [28, 49, 20], _i.e._ $|\tilde{\Delta}_{\rm
m}-\Delta_{\rm m}|\ll\Delta_{\rm m}$. Therefore, from now on we can assume
$\tilde{\Delta}_{\rm m}\simeq\Delta_{\rm m}$ and $\tilde{\Delta}_{\rm
fb}\simeq\Delta_{\rm fb}$. The explicit expression of the steady-state value
$m_{\rm s}$ is given by

$m_{\rm s}=\frac{E(\gamma_{\rm fb}+{\rm i}\Delta_{\rm
fb})}{g^{2}+\\{\gamma_{\rm m}+{\rm i}\Delta_{\rm m}\\}\\{\gamma_{\rm fb}+{\rm
i}\Delta_{\rm fb}\\}},$ (6)

which is generally complex, and its phase can be adjusted by varying the phase
of the driving field. Then, the corresponding linearized QLEs describing the
dynamics of the quantum fluctuations are given by (for simplicity, we use
$\mathcal{O}$ to denote fluctuation operators $\delta\mathcal{O}$)

$\displaystyle\delta\dot{a}$ $\displaystyle=$ $\displaystyle-(\gamma_{\rm
fb}+{\rm i}\tilde{\Delta}_{\rm fb})\delta a-{\rm i}G_{\rm a}(\delta d+\delta
d^{\dagger})-{\rm i}gm+\sqrt{2\gamma_{\rm a}}\delta a_{\rm fb}^{\rm in},$
$\displaystyle\delta\dot{b}$ $\displaystyle=$ $\displaystyle-(\gamma_{\rm
b}+{\rm i}\omega_{\rm b})\delta b-{{\rm i}(G_{\rm m}^{*}\delta m+G_{\rm
m}\delta m^{\dagger})}+\sqrt{2\gamma_{b}}\delta b_{\rm in},$
$\displaystyle\delta\dot{m}$ $\displaystyle=$ $\displaystyle-(\gamma_{\rm
m}+{\rm i}\Delta_{\rm m})\delta m-{\rm i}g\delta a-{\rm i}G_{\rm m}(\delta
b+\delta b^{\dagger})+\sqrt{2\gamma_{\rm m}}\delta m_{\rm in},$
$\displaystyle\delta\dot{d}$ $\displaystyle=$ $\displaystyle-(\gamma_{\rm
d}+{\rm i}\omega_{\rm d})\delta d-{{\rm i}G_{\rm a}(\delta a+\delta
a^{\dagger})}+\sqrt{2\gamma_{d}}\delta d_{\rm in},$ (7)

where $G_{\rm a}=g_{\rm a}a_{\rm s}$ represents the effective optomechanical
coupling strength, and $G_{\rm m}=g_{\rm m}m_{\rm s}$ denotes the effective
magnomechanical coupling strength. The QLEs (II) are linear with respect to
annihilation and creation operators. Therefore, it is convenient to introduce
the quadrature fluctuations, denoted as $\delta q$ and $\delta p$, for the
cavity, magnon and two mechanical modes, which are defined as
$\delta{\mathcal{O}}=(\delta q_{\mathcal{O}}+{\rm i}\delta
p_{\mathcal{O}})/\sqrt{2}$ with ${\rm\mathcal{O}}\in\\{a,b,d,m\\}$, satisfy
the commutation relation $[\delta q_{\mathcal{O}},\delta p_{\mathcal{O}}]={\rm
i}$ (a similar definition applies for corresponding input noise operators
$q_{\mathcal{O}}^{\rm in}$ and $p_{\mathcal{O}}^{\rm in}$ ). Subsequently, the
linearized QLEs describing the quadrature fluctuations $(\delta q_{\rm
d},\delta p_{\rm d},\delta q_{\rm a},\delta p_{\rm a},\delta q_{\rm m},\delta
p_{\rm m},\delta q_{\rm b},\delta p_{\rm b})$ can be expressed in a compact
matrix form as

$\dot{\mathcal{R}}(t)=A\mathcal{R}(t)+\mathcal{D}\mathcal{R}_{\rm in}(t),$ (8)

where $\mathcal{R}(t)=[\delta q_{\rm d}(t),\delta p_{\rm d}(t),\delta q_{\rm
a}(t),\delta p_{\rm a}(t),\delta q_{\rm m}(t),\delta p_{\rm m}(t),\delta
q_{\rm b}(t),\\\ \delta p_{\rm b}(t)]^{\top}$ (where $\top$ represents
_transposition_) is the vector of the quadrature fluctuations,
$\mathcal{R}_{\rm in}(t)=[q_{\rm d}^{\rm in}(t),p_{\rm d}^{\rm
in}(t),q_{a}^{\rm in}(t),p_{a}^{\rm in}(t),q_{\rm m}^{\rm in}(t),p_{\rm
m}^{in}(t),q_{\rm b}^{\rm in}(t),p_{\rm b}^{\rm in}(t)]^{\top}$ is the vector
for the corresponding input noise quadratureand $\mathcal{D}={\rm
diag}[\sqrt{2\gamma_{\rm d}},\sqrt{2\gamma_{\rm d}},\sqrt{2\gamma_{\rm
a}},\sqrt{2\gamma_{a}}\sqrt{2\gamma_{\rm m}},\sqrt{2\gamma_{\rm
m}},\sqrt{2\gamma_{\rm b}},\sqrt{2\gamma_{\rm b}}]$. Here, $A$ is the drift
matrix which is given by

$A=\begin{pmatrix}-\gamma_{\rm d}&\omega_{\rm d}&0&0&0&0&0&0\\\ -\omega_{\rm
d}&-\gamma_{\rm d}&-G_{\rm a}&0&0&0&0&0\\\ 0&0&-\gamma_{\rm fb}&\Delta_{\rm
fb}&0&g&0&0\\\ -G_{a}&0&-\Delta_{fb}&-\gamma_{\rm fb}&-g&0&0&0\\\
0&0&0&g&-\gamma_{\rm m}&\Delta_{\rm m}&0&0\\\ 0&0&-g&0&-\Delta_{\rm
m}&-\gamma_{\rm m}&-G_{\rm m}&0\\\ 0&0&0&g&0&0&-\gamma_{\rm b}&\omega_{b}\\\
0&0&0&0&0&G_{\rm m}&-\omega_{\rm b}&-\gamma_{\rm b}\end{pmatrix}.$ (9)

The system is stable if the real parts of all the eigenvalues of the drift
matrix $A$ are negative. This corresponds to the so-called Routh-Hurwitz
criterion [52].

Figure 3: Density plot of the three bipartite entanglements (a) entanglement
of the moving end mirror with mechanical mode of the YIG sphere
$\mathcal{E}_{\rm db}$, (b) magnon mode $\mathcal{E}_{\rm dm}$ and (c)
$\mathcal{E}_{\rm da}$ as a function of normalized detunings $\Delta_{\rm
m}/\omega_{\rm b}$ and $\Delta_{\rm a}/\omega_{\rm b}$ in the presence of
coherent feedback with reflection coefficient $\tau=0.9$ and phase $\phi=\pi$.
All other parameters are the same as in Fig.2.

##  III STEADY STATE ENTANGLEMENT

In this section, we explore three stationary bipartite entanglements: between
the mechanical resonator and the mechanical mode of the magnon YIG sphere, the
mechanical resonator and the magnon mode, and the mechanical resonator and the
cavity mode. Since the quantum noises are white in nature and the dynamics is
linearized, the state of the system will be a continuous variable (CV)
Gaussian state. Therefore, the steady state of the system is completely
characterized by an $8\times 8$ covariance matrix (CM) $\sigma$, defined as
$\sigma_{\rm ij}(t)=\langle\mathcal{R}_{\rm i}(t)\mathcal{R}_{\rm
j}(t^{\prime})+\mathcal{R}_{\rm j}(t^{\prime})\mathcal{R}_{\rm
i}(t)\rangle/2$. Then, the steady state of the $\sigma$ can be obtained by
solving directly the Lyapunov equation [6]:

$A\sigma+\sigma A^{\top}=-\cal{B},$ (10)

where $\cal{B}$ is the diffusion matrix defined by $\mathcal{B}_{\rm
ij}\delta(t-t^{\prime})=\langle\mathcal{R}_{{\rm in}_{\rm
i}}(t)\mathcal{R}_{{\rm in}_{\rm i}}(t^{\prime})+\mathcal{R}_{{\rm in}_{\rm
i}}(t^{\prime})\mathcal{R}_{{\rm in}_{\rm i}}(t)\rangle/2$, reads as

$\displaystyle\mathcal{B}$ $\displaystyle=$
$\displaystyle\text{diag}[\gamma_{\rm d}{\cal N}_{\rm d},\gamma_{\rm d}{\cal
N}_{\rm d},\gamma_{\rm a_{eff}}{\cal N}_{\rm a},\gamma_{\rm a_{\rm eff}}{\cal
N}_{\rm a},\gamma_{\rm m}{\cal N}_{\rm m},\gamma_{\rm m}{\cal N}_{\rm
m},\gamma_{b}{\cal N}_{\rm b},$ (11) $\displaystyle\gamma_{\rm b}{\cal N}_{\rm
b}],$

where $\gamma_{\rm a_{eff}}=\gamma_{\rm a}t^{2}\\{1-\tau\cos(\phi)\\}^{2}$ and
${\cal N}_{\cal O}=2N_{\cal O}+1$ with ${\cal O}=$d, a, m and b.

###  III.1 Stationary Bipartite Entanglement

The entanglement between any two of the four bipartite states can be
quantified by tracing out two of the four modes and considering the
logarithmic negativity, $\mathcal{E}_{\rm N}$ [47, 48]. We denote logarithmic
negativity for the moving end mirror-mechanical mode of the magnon, magnon-
moving end mirror and moving end mirror-cavity mode bimodal parition as
$\mathcal{E}_{\rm db}$, $\mathcal{E}_{\rm dm}$ and $\mathcal{E}_{\rm da}$,
respectively. The reduced state is still Gaussian and fully characterized by
its $4\times 4$ covariance matrix $\Sigma_{\rm i}$ where $\rm i\in\\{\rm
db,dm,da\\}$.Then the covariance matrix $\Sigma_{\rm i}$ associated with any
of the two modes is given by

$\Sigma_{\rm i}=\begin{pmatrix}\mathcal{X}_{\rm i}&\mathcal{Z}_{\rm i}\\\
\mathcal{Z}^{T}_{\rm i}&\mathcal{Y}_{\rm i}\end{pmatrix},$ (12)

where $\mathcal{X}_{\rm i}$ and $\mathcal{Y}_{\rm i}$ are $2\times 2$ sub-
matrices describing the auto-correlations between the two modes. The $2\times
2$ sub-matrix $\mathcal{Z}_{\rm i}$ in Eq. (12) denotes the cross-correlations
between the two modes. The logarithmic negativity $\mathcal{E}_{\rm i}$ is a
measure of the entanglement in the bipartite subsystem in the CV system, it
can be written as [45, 46, 47, 48]

$\mathcal{E}_{\rm i}=\max\left(0,-\log(2\nu^{-}_{\rm i})\right),$ (13)

with $\nu^{-}_{\rm i}$ being the smallest symplectic eigenvalue of the partial
transposed covariance matrix $\Sigma_{\rm i}$ of the two modes

$\nu^{-}_{\rm i}=2^{\frac{-1}{2}}\left(\Gamma_{\rm i}-\sqrt{\Gamma^{2}_{\rm
i}-4\det\Sigma_{\rm i}}\right)^{\frac{1}{2}},$ (14)

where $\Gamma_{\rm i}=\det\mathcal{X}_{\rm i}+\det\mathcal{Y}_{\rm
i}-2\det\mathcal{Z}_{\rm i}$. The two modes are entangled if the condition
$\nu^{-}_{\rm i}<1/2$ (_i.e._ when $\mathcal{E}_{\rm i}>0$) is satisfied.

We discuss steady state quantum correlations between two and three modes under
various effects by taking in consideration experimental values reported in
[53]: $\omega_{\rm a}/2\pi=10$ GHz, $\omega_{\rm b}/2\pi=10\,\unit{MHz}$,
$\gamma_{\rm a}/2\pi=1\unit{MHz}$, $\gamma_{\rm b}/2\pi=100\,\unit{Hz}$,
$\gamma_{\rm m}/2\pi=1$ MHz, $g/2\pi=3.2$ MHz, $G_{\rm
m}/2\pi=\,4.8\unit{MHz}$, considering $\omega_{\rm d}/2\pi=10\unit{MHz}$,
$\gamma_{\rm d}/2\pi=100\unit{Hz}$ and $G_{\rm a}/2\pi=3.2$ MHz and $T_{\rm
a}=T_{\rm m}=10\,\unit{mK}$ is the ambient temperature for the magnon and
microwave cavity modes which are assumed to be equal.

In Fig. 2 we plot the stationary entanglement between the movable mirror and
the mechanical mode of the YIG sphere Fig. 2(a)), the movable mirror and the
magnon mode (Fig. 2(b)), and the movable mirror and the photon (Fig.2(c)).
These entanglements are measured by using the logarithmic negativities
$\mathcal{E}_{\rm db}$, $\mathcal{E}_{\rm dm}$, and $\mathcal{E}_{\rm da}$,
respectively. In Fig. 2, we plot $\mathcal{E}_{\rm db}$ as a function of the
reflection coefficient $\tau$ and phase $\phi/\pi$ for fixed values of the
detunings $\Delta_{\rm a}=\Delta_{\rm a}=\omega_{\rm b}$. The entanglement
measure between movable end mirror and mechanical mode of the YIG sphere
increases with $\tau$ and reaches its maximum value for $\phi\approx\pi$ and
reflection coefficient $\tau\approx 1.0$. Similarly, the stationary
entanglement of the movable mirror with the magnon and cavity modes is
measured using the logarithmic negativity $\mathcal{E}_{\rm dm}$ and
$\mathcal{E}_{\rm da}$ as a functions of $\tau$ and $\phi/\pi$, as shown in
Figs. 2(b) and 2(c), respectively. The features observed in the entanglement
of the movable mirror with the magnon and cavity modes are similar to those
present in $\mathcal{E}_{\rm db}$. It is evident from Fig. 2 that the
entanglement between different bimodes can be enhanced with a suitable choice
of phase $\phi$ and reflection coefficient $\tau$. The three bipartite
entanglements reach their maximum value when $\gamma_{\rm fb}=\gamma_{\rm
a}+2\tau\gamma_{\rm a}$ and $\Delta_{\rm fb}=\Delta_{\rm a}$, _i.e._ when
$\phi=\pi$.

In the presence of coherent feedback, the entanglement of the moving end
mirror with mechanical mode of the YIG sphere $\mathcal{E}_{\rm db}$, magnon
$\mathcal{E}_{\rm db}$ and cavity $\mathcal{E}_{\rm db}$ modes as a function
of normalized detunings $\Delta_{\rm m}/\omega_{\rm b}$ and $\Delta_{\rm
a}/\omega_{\rm b}$ are shown Fig. 3. It is clear from Fig. 3(a), Fig. 3(b),
and Fig. 3(c) that the three bipartite entanglements, $\mathcal{E}_{\rm db}$,
$\mathcal{E}_{\rm dm}$, and $\mathcal{E}_{\rm da}$, in the presence of
coherent feedback exist only within a small interval of values for
$\Delta_{\rm a}$, with the maximum occurring around $\Delta_{\rm
a}\approx\omega_{\rm b}$.

Figure 4: Plot of three bipartite entanglements: $\mathcal{E}_{\rm db}$ (black
curves), $\mathcal{E}_{\rm dm}$ (blue curves), and $\mathcal{E}_{\rm da}$ (red
curves) as functions of the temperature T (K) for different values of the
reflection coefficient $\tau=0.6$ (solid curves) and $\tau=0.7$ (dashed
curves), with a fixed value of phase $\phi=\pi$ and detuning $\Delta_{\rm
a}=\Delta_{\rm a}=\omega_{\rm b}$. All the others parameters are the same as
in Fig.2.

It is important to study the robustness of the entanglement with respect to
the environmental temperature $T$, _i.e._ $T=T_{\rm b}=T_{\rm d}$. In Fig.4 we
plot three bipartite entanglements: $\mathcal{E}_{\rm db}$ (black curves),
$\mathcal{E}_{\rm dm}$ (blue curves), and $\mathcal{E}_{\rm da}$ (red curves)
against the environmental temperature $T$ for $\tau=0.7$ (solid curves) and
$\tau=0.9$ (dashed curves) when $\Delta_{\rm a}=\Delta_{\rm m}=\omega_{\rm b}$
and the phase $\phi=\pi$. For $\tau=0.7$, a significant amount of the
entanglement between the moving end mirror and the mechanical mode of the YIG
sphere, moving end mirror and magnon mode, and moving end mirror with cavity
mode are present up to temperatures $T$ (approximately) of $0.3\,\unit{K}$,
$0.9\,\unit{K}$, and $1.2\,\unit{K}$, as shown in Fig. 4 (solid curves).
However, the three bipartite entanglements $\mathcal{E}_{\rm db}$,
$\mathcal{E}_{\rm dm}$, and $\mathcal{E}_{\rm da}$ are more robust against the
environmental temperature $T$ and persist up to $0.7\,\unit{K}$ ,
$2.2\,\unit{K}$, and $3.1\,\unit{K}$ when the reflection coefficient is fixed
at $\tau=0.9$, as shown in Fig. 4 (dashed curves).

Figure 5: (a) Residual contangle $\mathcal{R}_{\rm min}$ as a function of
normalized magnon detuning $\Delta_{\rm a}/\omega_{\rm b}$ and cavity detuning
$\Delta_{\rm m}/\omega_{\rm b}$ for the fixed values of the phase
$\phi=0.65\pi$ and the reflection coefficient $\tau=0.7$. (b)
$\mathcal{R}_{\rm min}$ as a function of reflection coefficient $\tau$ and
phase $\phi$ for a constant value of the cavity and the magnon detunings
$\Delta_{\rm a}=\Delta_{\rm m}=0.5\omega_{\rm b}$. (c) $\mathcal{R}_{\rm min}$
as a function of the environmental temperature T(K), _i.e_ , ${\rm T}={\rm
T}_{\rm b}={\rm T}_{\rm d}$ for $\tau=0.5$ (black curve), $\tau=0.6$ (blue
dashed curve) and $\tau=0.7$ (red dashed curve) with $\Delta_{\rm
a}=\Delta_{\rm m}=0.5\,\omega_{\rm b}$ and $\phi=0.65\,\pi$. (d)
$\mathcal{R}_{\rm min}$ as function of phase $\phi/\pi$ for $\tau=0.5$ (black
curve), $\tau=0.6$ (blue dashed curve) and $\tau=0.7$ (red dashed curve) with
$\Delta_{\rm a}=\Delta_{\rm m}=0.5\,\omega_{\rm b}$ at fixed value of ${\rm
T}=100\,\unit{mK}$ and $\Delta_{\rm a}=\Delta_{\rm m}=0.5\omega_{\rm b}$. All
the others parameters are the same as in Fig.2.

###  III.2 Tripartite Entanglement

In this section we discuss the genuine tripartite entanglement and use the
residual contangle to examine tripartite movable mirror-magnon-mechanical mode
of YIG sphere entanglement [54], defined as

$\mathbb{R}^{\rm i|\rm jk}\equiv{\cal E}_{\rm i|\rm jk}-{\cal E}_{\rm i|\rm
j}-{\cal E}_{\rm i|\rm k},\quad{\rm i,j,k\in\\{d,m,b\\}}$ (15)

where ${\cal E}_{x|y}$ is the contangle of two modes $x$ and $y$ ($y$ composed
of one or two modes), and is quantified as the squared logarithmic negativity.
For Gaussian states, the minimum residual contangle offers an exact
measurement of tripartite entanglement when $\mathbb{R}^{\rm i|\rm jk}>0$

${\cal R}_{\rm min}\equiv{\rm min}\left(\mathbb{R}^{\rm d|\rm
mb},\,\mathbb{R}^{\rm m|\rm db},\,\mathbb{R}^{\rm b|\rm dm}\right).$ (16)

The quantum entanglement’s monogamy is satisfied by the residual contangle, is
similar to the Coffman-Kundu-Wootters monogamy inequality [55] holding for the
system of three qubits. Hence, a genuine tripartite entanglement emerges only
when the minimum residual entanglement of the system ${\cal R}_{\rm min}$ is
greater then _zero_ ,_i.e_. ${\cal R}_{\rm min}\,{>}\,0$.

In Fig.5, we investigate the minimum value of the residual contangle
$\mathcal{R}_{\rm min}$ given in Eq.(16), concerning various parameters in the
system. Fig.5(a) shows dependence of the contangle $\mathcal{R}_{\rm min}$ on
the normalized detunings $\Delta_{\rm a}/\omega_{\rm b}$ and $\Delta_{\rm
m}/\omega_{\rm b}$ for $\tau=0.75$ and phase $\phi=\pi$. It is evident from
Fig. 5(a) that $\mathcal{R}_{\rm min}$ exists only within a limited range of
values for $\Delta_{\rm a}$ and $\Delta_{\rm m}$, centered around $\Delta_{\rm
m}=\Delta_{\rm a}\approx 0.5\omega_{\rm b}$. Fig.5(b), shows the dependence of
$\mathcal{R}_{\rm min}$ on feedback parameters reflection coefficient $\tau$
and $\phi/\pi$ for fixed values of the detunings $\Delta_{\rm a}=\Delta_{\rm
m}=0.5\omega_{\rm b}$. It is clear that the optimal values for the reflection
coefficient and phase for the contangle $\mathcal{R}_{\rm min}$ is correspond
to red region in Fig.5(b) for $\Delta_{\rm a}=\Delta_{\rm m}=0.5\omega_{\rm
b}$.

We further study the effects of mechanical thermal noise on the steady state
residual contangle $\mathcal{R}_{\rm min}$ as a function of different
parameters of the system. In Fig.5(c), we plot the $\mathcal{R}_{\rm min}$ as
a function temperature T (K) of the two phononic modes, _i.e._ $T$=$T_{\rm
b}$=$T_{\rm d}$ for different values of the reflection coefficient $\tau=0.5$
(black curve), $0.6$ (blue curve) and $0.7$ (red curve) at fixed values of the
phase $\phi=0.65\pi$ and of the detuning $\Delta_{\rm a}=\Delta_{\rm
m}=0.5\omega_{\rm b}$. It is noted that $\mathcal{R}_{\rm min}$ is very
sensitive to the temperature of mechanical modes and quickly decays for large
values of the temperature. In high temperatures, thermal fluctuations always
suppress the residual contangle. In Fig.5(d), we plot $\mathcal{R}_{\rm min}$
as a function of phase $\phi/\pi$ for different values of reflection
coefficient $\tau=0.5$ (black curve), $0.6$ (blue curve) and $0.7$ (red curve)
when detuning is fixed at $\Delta_{\rm a}=\Delta_{\rm m}=0.5\omega_{\rm b}$.
One can easily observe that the stationary residual contangle
$\mathcal{R}_{\rm min}$ reaches it maximum value at phase $\phi=0.65\,\pi$.

##  IV Conclusions

We have introduced a theoretical scheme for generating stationary bipartite
and genuine tripartite entanglement in cavity magnomechanical systems with a
moving end mirror through coherent feedback. Our study illustrates that the
resultant bipartite and genuine tripartite quantum correlations can be
controlled and significantly improved through the implementation of coherent
feedback. Moreover, we have demonstrated the robustness of the resulting
quantum entanglement in the presence of environmental temperature associated
with phononic modes. We have investigated that the cavity magnomechanical
systems could provide a promising platform for the study of macroscopic
quantum phenomena just by implementing the appropriate feedback control. The
proposed scheme to enhance quantum correlations through coherent feedback can
be of interest for applications in quantum information processing.

## Acknowledgements

M. Asjad and D. Dutykh have been supported by the Khalifa University of
Science and Technology under Award No. FSU-2023-014).

## References

  * Bennett  _et al._ [1993] C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels, Phys. Rev. Lett. 70, 1895 (1993).
  * Bennett and Wiesner [1992] C. H. Bennett and S. J. Wiesner, Communication via one- and two-particle operators on einstein-podolsky-rosen states, Phys. Rev. Lett. 69, 2881 (1992).
  * Scarani  _et al._ [2005] V. Scarani, S. Iblisdir, N. Gisin, and A. Acín, Quantum cloning, Rev. Mod. Phys. 77, 1225 (2005).
  * Ekert [1991] A. K. Ekert, Quantum cryptography based on bell’s theorem, Phys. Rev. Lett. 67, 661 (1991).
  * Palomaki  _et al._ [2013] T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, Entangling mechanical motion with microwave fields, Science 342, 710 (2013).
  * Vitali  _et al._ [2007] D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, Optomechanical entanglement between a movable mirror and a cavity field, Phys. Rev. Lett. 98, 030405 (2007).
  * Teufel  _et al._ [2011] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds, Sideband cooling of micromechanical motion to the quantum ground state, Nature 475, 359 (2011).
  * O’Connell  _et al._ [2010] A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. T. Sank, H. Wang, M. P. Weides, J. Wenner, J. M. Martinis, and A. N. Cleland, Quantum ground state and single-phonon control of a mechanical resonator, Nature 464, 697 (2010).
  * Barzanjeh  _et al._ [2012] S. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, Reversible optical-to-microwave quantum interface, Phys. Rev. Lett. 109, 130503 (2012).
  * Abdi  _et al._ [2012] M. Abdi, S. Pirandola, P. Tombesi, and D. Vitali, Entanglement swapping with local certification: Application to remote micromechanical resonators, Phys. Rev. Lett. 109, 143601 (2012).
  * Manninen  _et al._ [2018] J. Manninen, M. Asjad, R. Ojajärvi, P. Kuusela, and F. Massel, Clauser-horne-shimony-holt bell inequality test in an optomechanical device, Phys. Rev. A 98, 043831 (2018).
  * Ockeloen-Korppi  _et al._ [2018] C. F. Ockeloen-Korppi, E. Damskägg, J.-M. Pirkkalainen, M. Asjad, A. A. Clerk, F. Massel, M. J. Woolley, and M. A. Sillanpää, Stabilized entanglement of massive mechanical oscillators, Nature 556, 478 (2018).
  * Riedinger  _et al._ [2018] R. Riedinger, A. Wallucks, I. Marinković, C. Löschnauer, M. Aspelmeyer, S. Hong, and S. Gröblacher, Remote quantum entanglement between two micromechanical oscillators, Nature 556, 473 (2018).
  * Xiong  _et al._ [2017a] H. Xiong, L.-G. Si, and Y. Wu, Precision measurement of electrical charges in an optomechanical system beyond linearized dynamics, Applied Physics Letters 110, 171102 (2017a).
  * Xiong  _et al._ [2017b] H. Xiong, Z.-X. Liu, and Y. Wu, Highly sensitive optical sensor for precision measurement of electrical charges based on optomechanically induced difference-sideband generation, Opt. Lett. 42, 3630 (2017b).
  * Caves [1980] C. M. Caves, Quantum-mechanical radiation-pressure fluctuations in an interferometer, Phys. Rev. Lett. 45, 75 (1980).
  * Abramovici  _et al._ [1992] A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, Ligo: The laser interferometer gravitational-wave observatory, Science 256, 325 (1992).
  * Braginsky and Vyatchanin [2002] V. Braginsky and S. Vyatchanin, Low quantum noise tranquilizer for fabry–perot interferometer, Physics Letters A 293, 228 (2002).
  * Bloch [1930] F. Bloch, Zur Theorie des Ferromagnetismus, Zeitschrift für Physik 61, 206 (1930).
  * Lachance-Quirion  _et al._ [2019] D. Lachance-Quirion, Y. Tabuchi, A. Gloppe, K. Usami, and Y. Nakamura, Hybrid quantum systems based on magnonics, Applied Physics Express 12, 070101 (2019).
  * Yuan  _et al._ [2022] H. Yuan, Y. Cao, A. Kamra, R. A. Duine, and P. Yan, Quantum magnonics: When magnon spintronics meets quantum information science, Physics Reports 965, 1 (2022), quantum magnonics: When magnon spintronics meets quantum information science.
  * Li  _et al._ [2018] J. Li, S.-Y. Zhu, and G. S. Agarwal, Magnon-photon-phonon entanglement in cavity magnomechanics, Phys. Rev. Lett. 121, 203601 (2018).
  * Asjad  _et al._ [2023] M. Asjad, J. Li, S.-Y. Zhu, and J. You, Magnon squeezing enhanced ground-state cooling in cavity magnomechanics, Fundamental Research 3, 3 (2023).
  * Osada  _et al._ [2018] A. Osada, A. Gloppe, R. Hisatomi, A. Noguchi, R. Yamazaki, M. Nomura, Y. Nakamura, and K. Usami, Brillouin light scattering by magnetic quasivortices in cavity optomagnonics, Phys. Rev. Lett. 120, 133602 (2018).
  * Osada  _et al._ [2016] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, and Y. Nakamura, Cavity optomagnonics with spin-orbit coupled photons, Phys. Rev. Lett. 116, 223601 (2016).
  * Zhang  _et al._ [2016a] X. Zhang, N. Zhu, C.-L. Zou, and H. X. Tang, Optomagnonic whispering gallery microresonators, Phys. Rev. Lett. 117, 123605 (2016a).
  * Fan  _et al._ [2022] Z.-Y. Fan, R.-C. Shen, Y.-P. Wang, J. Li, and J. Q. You, Optical sensing of magnons via the magnetoelastic displacement, Phys. Rev. A 105, 033507 (2022).
  * Zhang  _et al._ [2016b] X. Zhang, C.-L. Zou, L. Jiang, and H. X. Tang, Cavity magnomechanics, Science Advances 2, e1501286 (2016b).
  * Wiseman and Milburn [1993a] H. M. Wiseman and G. J. Milburn, Quantum theory of optical feedback via homodyne detection, Phys. Rev. Lett. 70, 548 (1993a).
  * Wiseman and Milburn [1993b] H. M. Wiseman and G. J. Milburn, Quantum theory of field-quadrature measurements, Phys. Rev. A 47, 642 (1993b).
  * Mancini  _et al._ [1998] S. Mancini, D. Vitali, and P. Tombesi, Optomechanical cooling of a macroscopic oscillator by homodyne feedback, Phys. Rev. Lett. 80, 688 (1998).
  * Courty  _et al._ [2001] J.-M. Courty, A. Heidmann, and M. Pinard, Quantum limits of cold damping with optomechanical coupling, The European Physical Journal D 17, 399 (2001).
  * Vitali  _et al._ [2002] D. Vitali, S. Mancini, L. Ribichini, and P. Tombesi, Mirror quiescence and high-sensitivity position measurements with feedback, Phys. Rev. A 65, 063803 (2002).
  * Ristè  _et al._ [2013] D. Ristè, M. Dukalski, C. A. Watson, G. de Lange, M. J. Tiggelman, Y. M. Blanter, K. W. Lehnert, R. N. Schouten, and L. DiCarlo, Deterministic entanglement of superconducting qubits by parity measurement and feedback, Nature 502, 350 (2013).
  * Rossi  _et al._ [2017] M. Rossi, N. Kralj, S. Zippilli, R. Natali, A. Borrielli, G. Pandraud, E. Serra, G. Di Giuseppe, and D. Vitali, Enhancing sideband cooling by feedback-controlled light, Phys. Rev. Lett. 119, 123603 (2017).
  * Clark  _et al._ [2017] J. B. Clark, F. Lecocq, R. W. Simmonds, J. Aumentado, and J. D. Teufel, Sideband cooling beyond the quantum backaction limit with squeezed light, Nature 541, 191 (2017).
  * Kralj  _et al._ [2017] N. Kralj, M. Rossi, S. Zippilli, R. Natali, A. Borrielli, G. Pandraud, E. Serra, G. D. Giuseppe, and D. Vitali, Enhancement of three-mode optomechanical interaction by feedback-controlled light, Quantum Science and Technology 2, 034014 (2017).
  * Rossi  _et al._ [2018] M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, and A. Schliesser, Measurement-based quantum control of mechanical motion, Nature 563, 53 (2018).
  * Bushev  _et al._ [2006] P. Bushev, D. Rotter, A. Wilson, F. m. c. Dubin, C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl, and P. Zoller, Feedback cooling of a single trapped ion, Phys. Rev. Lett. 96, 043003 (2006).
  * Amazioug  _et al._ [2020] M. Amazioug, B. Maroufi, and M. Daoud, Using coherent feedback loop for high quantum state transfer in optomechanics, Physics Letters A 384, 126705 (2020).
  * Li  _et al._ [2017] J. Li, G. Li, S. Zippilli, D. Vitali, and T. Zhang, Enhanced entanglement of two different mechanical resonators via coherent feedback, Phys. Rev. A 95, 043819 (2017).
  * Koch  _et al._ [2010] M. Koch, C. Sames, A. Kubanek, M. Apel, M. Balbach, A. Ourjoumtsev, P. W. H. Pinkse, and G. Rempe, Feedback cooling of a single neutral atom, Phys. Rev. Lett. 105, 173003 (2010).
  * Gieseler  _et al._ [2012] J. Gieseler, B. Deutsch, R. Quidant, and L. Novotny, Subkelvin parametric feedback cooling of a laser-trapped nanoparticle, Phys. Rev. Lett. 109, 103603 (2012).
  * Genoni  _et al._ [2015] M. G. Genoni, J. Zhang, J. Millen, P. F. Barker, and A. Serafini, Quantum cooling and squeezing of a levitating nanosphere via time-continuous measurements, New Journal of Physics 17, 073019 (2015).
  * Vidal and Werner [2002] G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  * Plenio [2005] M. B. Plenio, Publisher’s Note: Logarithmic Negativity: A full entanglement monotone that Is not convex [Phys. Rev. Lett. 95, 090503 (2005)], Phys. Rev. Lett. 95, 119902 (2005).
  * Adesso  _et al._ [2004] G. Adesso, A. Serafini, and F. Illuminati, Extremal entanglement and mixedness in continuous variable systems, Phys. Rev. A 70, 022318 (2004).
  * Simon [2000] R. Simon, Peres-horodecki separability criterion for continuous variable systems, Phys. Rev. Lett. 84, 2726 (2000).
  * Potts  _et al._ [2021] C. A. Potts, E. Varga, V. A. S. V. Bittencourt, S. V. Kusminskiy, and J. P. Davis, Dynamical backaction magnomechanics, Phys. Rev. X 11, 031053 (2021).
  * Giovannetti and Vitali [2001] V. Giovannetti and D. Vitali, Phase-noise measurement in a cavity with a movable mirror undergoing quantum brownian motion, Phys. Rev. A 63, 023812 (2001).
  * Walls and Milburn [2012] D. F. Walls and G. J. Milburn,  _Quantum Optics_ (Springer Berlin, Heidelberg, 2012).
  * DeJesus and Kaufman [1987] E. X. DeJesus and C. Kaufman, Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations, Phys. Rev. A 35, 5288 (1987).
  * Fan  _et al._ [2023] Z.-Y. Fan, H. Qian, X. Zuo, and J. Li, Entangling ferrimagnetic magnons with an atomic ensemble via optomagnomechanics, Phys. Rev. A 108, 023501 (2023).
  * Adesso and Illuminati [2007] G. Adesso and F. Illuminati, Entanglement in continuous-variable systems: recent advances and current perspectives, Journal of Physics A: Mathematical and Theoretical 40, 7821 (2007).
  * Coffman  _et al._ [2000] V. Coffman, J. Kundu, and W. K. Wootters, Distributed entanglement, Phys. Rev. A 61, 052306 (2000).